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Quizzes > Mathematics > Advanced Math

Test Your Integration Skills with Our Free Quiz

Ready to tackle integral calculus questions? Dive in and prove your mastery!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration features calculus symbols area volume shapes on golden yellow background for integration quiz

Use this quiz to practice integration and see where your calculus skills need a tune-up. Work through area, volume, and technique problems, then review more practice questions and explore definite integral applications to strengthen weak spots before an exam and build speed.

What is the indefinite integral of x^2 with respect to x?
2x + C
x^2 + C
3x + C
x^3/3 + C
By the power rule for integration, ?x^n dx = x^(n+1)/(n+1) + C. Here n = 2, so ?x^2 dx = x^3/3 + C. This rule applies for any real n ? -1.
Compute the indefinite integral ? cos(x) dx.
-cos(x) + C
cos(x) + C
sin(x) + C
-sin(x) + C
The derivative of sin(x) is cos(x), so the integral of cos(x) is sin(x) + C. Integrals reverse differentiation rules.
Find ? e^x dx.
e^x + C
x e^x + C
e^x + x + C
ln(e^x) + C
The function e^x is unique in that its derivative is itself. Thus, ?e^x dx = e^x + C. No substitution is needed. Wikipedia: Exponential Function
What is ? 5 dx?
5 + C
5x + C
ln(5x) + C
x^5 + C
Integrating a constant k with respect to x gives kx + C. Here k = 5, so ?5 dx = 5x + C. This is a direct application of ?a dx = ax + C.
Calculate ? sin(x) dx.
cos(x) + C
sin(x) + C
-sin(x) + C
-cos(x) + C
Since the derivative of cos(x) is -sin(x), the integral of sin(x) must be -cos(x) + C. Integration reverses the sign from differentiation.
Find the indefinite integral ?(2x + 3) dx.
x^2 + 3x + C
2x^2 + 3 + C
2x^2/2 + 3x + C
x^2 + x + C
Split the integral: ?2x dx = x^2, ?3 dx = 3x. So ?(2x+3)dx = x^2 + 3x + C. Combining constants yields the final form.
What is ?(1/x) dx?
1/(x^2) + C
x ln|x| - x + C
ln|x| + C
e^x + C
The integral of 1/x is the natural logarithm of the absolute value of x: ln|x| + C. This arises because the derivative of ln|x| is 1/x.
Compute ? sec^2(x) dx.
tan(x) + C
sin(x) + C
cot(x) + C
sec(x) + C
The derivative of tan(x) is sec^2(x), making ?sec^2(x) dx = tan(x) + C. This is a standard result in trigonometric integration.
Evaluate the definite integral ? from 0 to 1 of x dx.
1/3
1/2
2/3
1
The antiderivative of x is x^2/2. Evaluating from 0 to 1 gives (1^2/2) - (0) = 1/2. This is straightforward area under the line y=x.
Compute ? from 0 to ?/2 of sin(x) dx.
0
?/2
1
2
The antiderivative of sin(x) is -cos(x). Evaluating from 0 to ?/2 yields [-cos(?/2) + cos(0)] = [0 + 1] = 1. This is a classic integral.
Find ? x e^x dx.
x e^x - e^x + C
x e^x + C
e^x + C
e^x(x + 1) + C
Use integration by parts: let u = x, dv = e^x dx. Then du = dx, v = e^x. So ?x e^x dx = x e^x - ?e^x dx = x e^x - e^x + C.
Evaluate ?(2x)/(1+x^2) dx.
ln(1 + x^2) + C
x^2/(1+x^2) + C
1/(1+x^2) + C
2 ln|x| + C
Use substitution u = 1 + x^2, du = 2x dx. The integral becomes ?du/u = ln|u| + C = ln(1 + x^2) + C.
Compute ? from 1 to 2 of 1/x dx.
ln(2) - 1
2
1/2
ln(2)
The antiderivative of 1/x is ln|x|. Evaluating from 1 to 2 gives ln(2) - ln(1) = ln(2).
Find ? 1/?x dx.
ln|x| + C
?x/2 + C
x^(3/2)/3 + C
2?x + C
Rewrite 1/?x as x^(-1/2). Integrate: x^(-1/2+1)/(-1/2+1) = x^(1/2)/(1/2) = 2?x + C.
Determine the area under y = x^2 from x = 0 to x = 2.
2
16/3
4
8/3
The definite integral of x^2 from 0 to 2 is [x^3/3]_0^2 = (8/3) - 0 = 8/3. This represents the area under the curve.
Compute ? ln(x) dx.
x ln(x) + C
x ln(x) - x + C
1/x + C
ln(x)/x + C
Use integration by parts: let u = ln(x), dv = dx. Then du = dx/x, v = x. So ?ln(x)dx = x ln(x) - ?1 dx = x ln(x) - x + C.
Evaluate ? from 0 to 1 of x e^(x^2) dx.
e^2/2 - 1/2
e^(1/2) - 1
e - 1
1/2 (e - 1)
Let u = x^2, du = 2x dx. Then ?x e^(x^2) dx = 1/2 ?e^u du = 1/2 e^u + C. From 0 to 1: 1/2(e^1 - e^0) = (e - 1)/2.
Find the volume generated by rotating y = ?x around the x-axis from x = 0 to 4.
? * ?0^4 x dx = 8?
? * ?0^4 x dx = 8?
? * ?0^4 x dx = 8?
? * ?0^4 x dx = 8?
Using the disk method: radius = y = ?x, area = ?x. Volume = ??0^4 x dx = ?[x^2/2]_0^4 = ?(16/2) = 8?.
Evaluate ? sin^2(x) dx.
-cos^2(x)/2 + C
x/2 + (sin(2x))/4 + C
x/2 - (sin(2x))/4 + C
sin(x) cos(x) + C
Use the identity sin^2(x) = (1 - cos(2x))/2. Integrate: ?sin^2(x)dx = ?(1/2)dx - ?(cos(2x)/2)dx = x/2 - (sin(2x))/4 + C.
Compute ? x^3/(x^2 + 1) dx.
x^2/2 - (1/2) ln(x^2+1) + C
ln(x^2+1) + C
x^2/2 + ln(x^2+1) + C
x^2 - ln(x^2+1) + C
Divide x^3 by x^2+1: x^3/(x^2+1) = x - x/(x^2+1). Integrate: ?x dx = x^2/2; ?x/(x^2+1)dx = (1/2)ln(x^2+1). Combine: x^2/2 - (1/2)ln(x^2+1) + C.
Evaluate ? e^(2x) sin(2x) dx.
e^(2x)/2 (sin(2x) - cos(2x)) + C
e^(2x) sin(2x) + C
e^(2x)/4 (sin(2x) + cos(2x)) + C
1/4 e^(2x) sin(2x) + C
Use integration by parts twice or complex exponentials. The result simplifies to (e^(2x)/2)(sin(2x) - cos(2x)) + C.
Find ? arctan(x) dx.
x arctan(x) + C
arctan(x)^2/2 + C
x arctan(x) - (1/2) ln(1+x^2) + C
ln(1+x^2) + C
Integration by parts: let u = arctan(x), dv = dx. Then du = dx/(1+x^2), v = x. So ?arctan(x)dx = x arctan(x) - ?x/(1+x^2)dx = x arctan(x) - (1/2)ln(1+x^2) + C.
Compute the improper integral ? from 0 to ? of x e^(-x^2) dx.
1/2
??/2
0
1
Let u = x^2, du = 2x dx. Then ? x e^(-x^2) dx = -1/2 ? e^(-u) du. Evaluating from 0 to ? gives 1/2.
Evaluate ? from 0 to ?/2 of ln(sin(x)) dx.
?/2
-? ln(2)
0
-(?/2) ln(2)
This classic integral equals - (?/2)ln(2). It can be shown using the Beta function or Fourier series.
0
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Study Outcomes

  1. Understand Fundamental Integration Concepts -

    Explain the meaning of antiderivatives and basic rules for integrating general functions in calculus.

  2. Apply Standard Integration Techniques -

    Solve integral calculus questions using methods like substitution, integration by parts, and partial fractions.

  3. Solve Definite Integrals Accurately -

    Compute exact values of definite integrals to find areas under curves and practice precise definite integral evaluation.

  4. Analyze Area and Volume Problems -

    Set up and evaluate integrals to determine the area and volume of complex geometric shapes.

  5. Interpret Quiz Feedback for Improvement -

    Identify common errors in integration problems quiz responses and develop strategies to enhance your skills.

  6. Choose Optimal Integration Methods -

    Compare various integration techniques and select the most efficient approach for different problem types.

Cheat Sheet

  1. Fundamental Theorem of Calculus -

    When tackling questions about integration, master the Fundamental Theorem of Calculus that links definite integrals and antiderivatives. In Part 1, F(x)=∫_a^x f(t)dt shows how differentiation reverses integration, while Part 2 gives ∫_a^b f′(x)dx=f(b)−f(a), letting you compute areas directly (Source: MIT OpenCourseWare). This tool is crucial on any integration problems quiz or integral calculus questions.

  2. u-Substitution Technique -

    u-Substitution is your go-to method for integrals where a composite function appears; set u equal to an inner function so du matches f ′(x)dx. For instance, ∫2x·cos(x^2+1)dx becomes ∫cos(u)du, yielding sin(u)+C (u=x^2+1). Practicing this on definite integral practice ensures you spot substitutions quickly.

  3. Integration by Parts and LIATE -

    Integration by parts comes from ∫u dv = u·v − ∫v du, and the LIATE mnemonic (Logs, Inverse trig, Algebraic, Trig, Exponential) helps choose u. For example, ∫x e^x dx = x e^x − ∫1·e^x dx = x e^x − e^x + C (Stewart's Calculus). This technique tackles tougher integral calculus questions and boosts your confidence on tricky integrals.

  4. Area and Volume Applications -

    Translating integrals into geometrical quantities is key: area under y=f(x) is ∫_a^b f(x)dx, while revolving around the x-axis gives volume V=π∫_a^b[f(x)]^2dx (Source: University of Cambridge). The washer method handles holes, and the shell method uses 2π∫ x·f(x)dx for cylindrical shells. Visualizing these setups turns abstract formulas into concrete problem-solving tools.

  5. Numerical Integration Methods -

    When antiderivatives are elusive in an integration techniques test, use numerical rules like Trapezoidal and Simpson's. Simpson's Rule, for instance, approximates ∫_a^b f(x)dx ≈ (Δx/3)[f(x_0)+4f(x_1)+f(x_2)] over each interval (Khan Academy). Regular practice on integration problems quizzes sharpens your error estimates and computational accuracy.

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