Integration Identities (2024)

Can you identify the antiderivative of these common integrands?

By notacountry

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19 Questions

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")$objective.click(() => {window.onClickObjective(reckoningQuestObjective.quest_number, reckoningQuestObjective.objective_number, reckoningQuestObjective.quest_id, reckoningQuestObjective.objective_id, $objective);});if (!window.mweb) {$j('#snark').hide();$objective.insertAfter($j('#gameOverMsg'));} else if (is_flagship_app_view) {$j('#snark').html();$objective.appendTo($j('#snark'));} else {$objective.appendTo($j('#reckoning .stats-wrapper'));}} else if (showSnark) {var snark = getSnark(userPct);var attachSnarkModalListener = false;if (snark !== '') {snark = ""+snark+"";if (!window.mweb) {if (false) {attachSnarkModalListener = true;snark += "";} else {snark += "";}}} else {$j('#snark').hide();}$j('#snark').html(snark);if(attachSnarkModalListener){ $j('#hide-snark-open-confirmation-modal').on('click', function () { var modalArgs = {title: 'Hide These Messages?',message: 'Are you sure you want to hide post-quiz messages?', onYes: function(remodal){ $j.ajax({ type: "GET", url: "/ajax/user_settings.php", data: {s: 0}, success: function(response) { if (response.success) { $j('#snark').hide(); remodal.close(); } else { if (response.responseJSON.message) { remodal.$modal.find('#confirmation-error').html(response.responseJSON.message); remodal.$modal.find('#confirmation-error').show(); } else { remodal.$modal.find('#confirmation-error').html('There was an error saving your preferences. Please contact feedback if you continue to see this issue.'); remodal.$modal.find('#confirmation-error').show(); } } }, error: function(response) { if (response.responseJSON.message) { remodal.$modal.find('#confirmation-error').html(response.responseJSON.message); remodal.$modal.find('#confirmation-error').show(); } else { remodal.$modal.find('#confirmation-error').html('There was an error saving your preferences. Please contact feedback if you continue to see this issue.'); remodal.$modal.find('#confirmation-error').show(); } } }); } }; SporcleLib.Modal._openConfirmationModal(modalArgs); }); }} else {$j('#snark').hide();}// Update get reckoning with playlist infovar playlistData;if (mweb) {playlistData = typeof window.playlist === 'object' && window.playlist.hasOwnProperty('games') ? window.playlist : null;} else {playlistData = window.Sporcle.gameData.playlist ? window.Sporcle.gameData.playlist.vm.data : null;}if (playlistData) {playlistReckoning(playlistData, userPct, false && !stopwatch);}$j('#playGameBar').addClass('reckoning');$j('#reckonBox').show();if (mweb) {/* mobile games only */$j('#reckoning-numright').html(numRight);$j('#reckoning-time').html(displayTime(timerSecs));if (stopwatch) {$j('#reckoning-time-title').text("Stopwatch");$j('#reckoning-time, #reckoning-score').addClass('timePast');}var checkToOpenFlorinPlayGoalModal = () => {if (userPct == 0) {return;}if (!florinsPlayGoalData) {return;}var openModal = false;if (florinsPlayGoalData.reward_claimable) {openModal = true;} else if (florinsPlayGoalData.curr_streak + 1 == florinsPlayGoalData.goal_days && florinsPlayGoalData.plays_today == 0) {// First play of the day, this should complete their play goalopenModal = true;}if (openModal) { window.SporcleLib.Modal.openFlorinPlayGoalModal({ // Fudge these values to be at the goal, since we haven't necessarily logged this play yet and it might be the 7th curr_streak : florinsPlayGoalData.goal_days, reward_claimable : true, plays_today : florinsPlayGoalData.plays_today + 1, goal_days : florinsPlayGoalData.goal_days, reward_amount : florinsPlayGoalData.reward_amount, })}}if (is_flagship_app_view) {let animations = window.animations;let hapticType = 'success';let completeMsg;let animConfig = {animationData : animations.checkmark,loop : true,playSegments : [[0, 30], [31, 75]]};if (userPct == 1) {animConfig.animationData = animations.perfectScore;animConfig.playSegments = [[0, 14], [15, 75]];completeMsg = "Perfect Score!";} else if (timerSecs == 0) {hapticType = 'error';animConfig.animationData = animations.outOfTime;animConfig.playSegments = [[0, 14], [15, 75]];completeMsg = "Out of Time!";} else if (_spks && !didGiveUp) {hapticType = 'error';animConfig.animationData = animations.bomb;animConfig.playSegments = [[0, 14], [15, 75]];completeMsg = "You Hit a Mine!";}window.SporcleLib.Modal.openModalFromTpl($j('#post-game-modal-tpl').html(), {show_loading : false,onInit : (modal) => {let $modal = modal.$modal; $modal.attr('id', 'postGameModal'); postGameAnim = lottie.loadAnimation({container : $modal.find('#postGameIcon')[0],animationData : animConfig.animationData,loop : animConfig.loop,autoplay : false});$modal.find('.modal-close').on('click', () => {window.continuePostQuiz();}); $modal.find('#completeMsg').html(completeMsg); $modal.find('#postPct').html(roundPercentScore(userPct));$modal.find('#postCorrect').html($j('#reckoning-score').text());if (snark) {$modal.find('#snark').html(snark);} else {$modal.find('#snark').hide();}setTimeout(() => {postGameAnim.playSegments(animConfig.playSegments, true);// if (hapticType) {// setTimeout(() => {// window.SporcleApp.api.playHaptic(hapticType);// }, 150);// }}, 150);},onClose : () => {checkToOpenFlorinPlayGoalModal();}});} else {checkToOpenFlorinPlayGoalModal();}}$j('#playGameBox').empty().hide();$j('#postGameBox').show();}function toggleshow() {showans = !showans;}function roundPercentScore(pctScore) {return _.round(100 * pctScore);}function updateBadge(userPct) {var earned = false;if (quizBadge && userPct > 0) { if (quizBadge.criteria && quizBadge.criteria.length) { var requiredCount = quizBadge.criteria.length; var earnedCount = 0; //Loop through possible badge criteria to see if we earned them quizBadge.criteria.forEach(function (criterion) { var comparison = parseInt(criterion.compare); if(criterion.name === 'percent'){ if ((comparison === 0 && userPct === parseFloat(criterion.value)) || //'0' means percent needs to equal (comparison === 1 && userPct > parseFloat(criterion.value)) || //'1' means percent needs to be greater than (comparison === -1 && userPct < parseFloat(criterion.value)) //'-1' means percent needs to be less than (at the time of writing, no badge exists like this) ) { earnedCount++; } } else if (criterion.name === 'month'){ var d = new Date(); //getMonth() starts at 0 if((d.getMonth() + 1 == criterion.value && comparison === 0) || (d.getMonth() + 1 > criterion.value && comparison === 1) || (d.getMonth() + 1 < criterion.value && comparison === -1) ){ earnedCount++; } } else if (criterion.name === 'day'){ var d = new Date(); if((d.getDate() == criterion.value && comparison === 0) || (d.getDate() > criterion.value && comparison === 1) || (d.getDate() < criterion.value && comparison === -1) ){ earnedCount++; } } else if (criterion.name === 'day_of_the_week'){ var d = new Date(); if((d.getDay() == criterion.value && comparison === 0) || (d.getDay() > criterion.value && comparison === 1) || (d.getDay() < criterion.value && comparison === -1) ){ earnedCount++; } } else if (criterion.name === 'avg_percent' && typeof averageScore !== 'undefined') {var roundedUserPct = _.round(userPct, 2); //matches backend precision in badge trackervar roundedAvgPct = _.round(averageScore, 2);if((roundedUserPct == roundedAvgPct && comparison === 0) || (roundedUserPct > roundedAvgPct && comparison === 1) || (roundedUserPct < roundedAvgPct && comparison === -1) ){ earnedCount++; } }}); earned = earnedCount == requiredCount;} else {earned = true;}} if (earned) {quizBadge.condition_earned = true;quizBadge.achieved_count++; var spliceIndex = null;_(quizBadge.remaining_conditions).each(function(condition, index) {if (quizBadge.condition_id == condition.condition_id) {spliceIndex = index;}});_(quizBadge.conditions).each(function(condition, index) {if (quizBadge.condition_id == condition.condition_id) {condition.earned = true;}});if (spliceIndex !== null) {quizBadge.remaining_conditions.splice(spliceIndex, 1); quizBadge.condition_earned_date = (new Date()).toLocaleDateString(undefined, { year: 'numeric', month: 'long', day: 'numeric' });; }if (quizBadge.remaining_conditions.length === 0 ) {if (quizBadge.badge_earned && quizBadge.levelable) {quizBadge.achieved_count = 0;quizBadge.level++;var currDate = new Date();quizBadge.last_level_date = window.SporcleLib.getMonthString(currDate.getMonth()) + ' ' + currDate.getDate() + ', ' + currDate.getFullYear();} else {quizBadge.badge_earned = true;var currDate = new Date();quizBadge.earned_date = window.SporcleLib.getMonthString(currDate.getMonth()) + ' ' + currDate.getDate() + ', ' + currDate.getFullYear();if (quizBadge.levelable) {quizBadge.level = 1;quizBadge.last_level_date = window.SporcleLib.getMonthString(currDate.getMonth()) + ' ' + currDate.getDate() + ', ' + currDate.getFullYear();quizBadge.achieved_count = 0;}}}if (badgeTooltip) {badgeTooltip.deactivate();}if (badgeTooltipRightRail) {badgeTooltipRightRail.deactivate();}var template = _.template($j('#quiz-badge-template').html());if (window.mweb) {$j('#main-wrapper .quiz-badge').replaceWith($j(template(quizBadge)));} else {quizBadge.tooltip_trigger_id = 'quiz-badge-art';$j('#main-quiz-content .quiz-badge').replaceWith($j(template(quizBadge)));quizBadge.tooltip_trigger_id = 'quiz-badge-art-right-rail';$j('#right-rail .quiz-badge').replaceWith($j(template(quizBadge)));if (!quizBadge.badge_earned) {if (window.setupUnearnedBadgeTooltip) {window.setupUnearnedBadgeTooltip($j('#quiz-badge-art'), quizBadge);window.setupUnearnedBadgeTooltip($j('#quiz-badge-art-right-rail'), quizBadge);} else {var conditionsTemp = _.template($j('#badge-conditions-template').html());var tooltipHtml = $j(conditionsTemp(quizBadge));const badgeTooltipOptions = { //this should match /sporcle-ui/tooltips/functionalTooltipstarget : true,tipJoint : 'bottom middle',showOn : 'mouseover',hideTriggers : ['target','tip'],stemLength : 8,stemBase : 12,hideOn : 'mouseout',removeElementsOnHide : true,background : 'white',borderColor : '#d8d8d8',borderWidth : 1,shadowOffset : [3,3],shadowBlur : 10,showEffect : 'slide',showEffectDuration : 0.1,hideEffectDuration : 0.3,className : 'badge-tooltip',shadowColor : 'rgba(0,0,0,0.25)'};badgeTooltip = new Opentip('#quiz-badge-art', tooltipHtml.html(), badgeTooltipOptions);badgeTooltipRightRail = new Opentip('#quiz-badge-art-right-rail', tooltipHtml.html(), badgeTooltipOptions);}} else {if (window.setupEarnedBadgeTooltip) {window.setupEarnedBadgeTooltip($j('#quiz-badge-art'), quizBadge);window.setupEarnedBadgeTooltip($j('#quiz-badge-art-right-rail'), quizBadge);}}}}}function showHintTooltip(x, y, hint) {$hint = $j('#hintTooltip');if ($hint.length == 0) {$hint = $j('

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').appendTo($j('body'));}if (hint) {var $afterContainer = $hint.find('.afterContainer');var $after = $hint.find('.after');// Set to max width before resize$afterContainer.css({width: '120px'});$hint.show();// Set tooltip text to hint$after.html(hint);// Get width of tooltip with textvar width = Math.ceil($after[0].offsetWidth) + 1;var leftShift = 0;if (mweb) {// Make sure the tooltip doesn't go off either side of the screenvar leftBound = 51;var rightBound = $j('body').width() - 71;var buf = 11;// Shift based on width of textvar shiftLimit = Math.max(width/2 - 20, 0);if (x < leftBound) {// Too close to left edge of screenleftShift = -shiftLimit;x = Math.max(x, buf);} else if (x > rightBound) {// Too close to right edge of screenleftShift = shiftLimit;x = Math.min(x, $j('body').width() - buf);}}// Set the position of the hint$hint.css({left: x + 'px',top: y + 'px',});// Set outer container size and position$afterContainer.css({width: width + 'px',left: '-' + ((width / 2) + leftShift) + 'px'});} else {$hint.hide();}}function getRandomColor() {var r = Math.round(Math.random() * 255);var g = Math.round(Math.random() * 255);var b = Math.round(Math.random() * 255);return 'rgb(' + r + ',' + g + ',' + b + ')';}function hintTooltipIsDisplayed() {return $j('#hintTooltip').is(':visible');}function getCurrentHintText() {return $j('#hintTooltip .afterContainer .after').text();}async function runStartCode(stopwatch) {if (typeof gamePageIsLoaded != "undefined" && gamePageIsLoaded) {if (window.startTriggered) {return;} else {window.startTriggered = true;}if (typeof stopwatch !== 'undefined') {if (stopwatch) {window.selectTimerSetting('stopwatch');} else {window.selectTimerSetting('timer');}}startGame(120);callSPHooks('postStart');}}function callSPHooks(name) {if (typeof _spHooks != 'undefined') {var hlen = _spHooks[name].length;for (var i = 0; i < hlen; i++) {_spHooks[name][i]();}}}

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02:00

f(x)∫f(x)dx
a
x^n
1/x
ae^(ax)
a^x
ln(x)
log_a(x)
sin(x)
cos(x)
tan(x)
f(x)∫f(x)dx
cot(x)
sec(x)
csc(x)
sec(x)tan(x)
csc(x)cot(x)
tan(x)^2
sinh(x)
cosh(x)
tanh(x)
Integration Identities (2024)

FAQs

Can you get different answers in integration? ›

On the other hand, there are no cases in which an integral actually has two different solutions; they can only "look" different. For example, x+c and x2+c cannot both be solutions to the same integral, because x and x2 don't differ by a constant.

Why is integration so much harder? ›

There are no general formulas for the integral of a product of functions, or for their composition, and that's why integration seems to be so much harder.

How to solve integration easily? ›

Integrals are solved various ways depending on the function being evaluated. The most basic way is to use the power rule. If the integral is definite then the answer will be a numerical value. However, if the integral is indefinite, then the answer will be another function.

Is integration tougher than differentiation? ›

Integration is generally much harder than differentiation. This little demo allows you to enter a function and then ask for the derivative or integral. You can also generate random functions of varying complexity. Differentiation is typically quite easy, taking a fraction of a second.

Can you learn integration without knowing differentiation? ›

Absolutely, studying integration before applications of derivatives is a viable approach. Integration involves finding antiderivatives and understanding the accumulation of quantities. It provides a conceptual framework for grasping the reverse process of differentiation.

How can I master integration? ›

Understand the basics of integral calculus:

Before attempting to master integration techniques, start with fundamental integration techniques such as the power rule, constant rule, and basic rules for integrating trigonometric and exponential functions.

What is Feynman's trick? ›

In practice, Feynman's trick is often applied to integrals with a single variable; the idea is to artificially introduce a second variable and then build a differential equation in terms of derivatives with respect to 1 Page 2 the new variable.

Is integration easy or not? ›

i.e., the integration is the reverse process of differentiation. But it is not possible (not easy) every time to apply the reverse process of differentiation to evaluate the integrals.

What is the easiest method of integration? ›

When you have to integrate rational functions, a method of integration using partial fractions is used. The reverse chain rule is also one of the easiest and most commonly used methods of integration.

How to study integration easily? ›

Integration Using Trigonometric Identities

In the integration of a function, if the integrand involves any kind of trigonometric function, then we use trigonometric identities to simplify the function that can be easily integrated. All these identities simplify integrand, that can be easily found out.

Can definite integrals have different solutions? ›

Definite Integrals have exactly one answer to a particular question as it is evaluated in a fixed limit. But Indefinite Integrals can have different solutions or answers to a particular question and it also depends on the constant which could be in exponential, logarithmic, trigonometric form etc.

Can differentiation have multiple answers? ›

Two different answers can be found when using implicit differentiation because the method involves using the chain rule, which can yield different results depending on the way it is applied. Additionally, implicit functions may have multiple solutions for the derivative, leading to different answers.

Are there multiple ways to solve an integral? ›

The different methods of integration include:
  • Integration by Substitution.
  • Integration by Parts.
  • Integration Using Trigonometric Identities.
  • Integration of Some particular function.
  • Integration by Partial Fraction.

Can math questions have multiple answers? ›

Yes, there are a lot of math problems that only have one answer. But, there are problems that can have more than one answer and all math problems have more than one way to get to the answer.

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