Integrals & Antiderivatives Practice Quiz with a Step-by-Step Interactive Lesson
Use the question set below to practice integrals and antiderivatives — the core skills behind area under a curve, accumulation, and many applications in Calculus. This lesson focuses on the most important integration tools you need early on: indefinite integrals \(\int f(x)\,dx\) as families of antiderivatives, the constant of integration \(+C\), the power rule for integration \(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\) (for \(n≠ -1\)), the special logarithmic case \(\int \dfrac{1}{x}\,dx=\ln|x|+C\), common exponential integrals like \(\int e^x\,dx=e^x+C\) and \(\int a^x\,dx=\dfrac{a^x}{\ln a}+C\), must-know trigonometric integrals like \(\int \sec^2 x\,dx=\tan x+C\) and \(\int \csc^2 x\,dx=-\cot x+C\), and quick pattern recognition for u-substitution (reverse chain rule), such as \(\int \dfrac{2x}{x^2+1}\,dx=\ln(x^2+1)+C\). You will also practice definite integrals and evaluation with the Fundamental Theorem of Calculus. If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
Answer the question set and review your mistakes at the end.
How this integrals and antiderivatives practice works
1. Take the practice set: answer the integrals and antiderivatives questions below.
2. Open the lesson (optional): review antiderivative rules, trig/exponential/log integrals, u-substitution patterns, and definite integrals.
3. Retry: return to the question set and apply the integration rules immediately.
What you will learn in the integrals & antiderivatives lesson
Indefinite integrals & the constant of integration
Antiderivative meaning: \(\int f(x)\,dx = F(x)+C\), where \(F'(x)=f(x)\)
+C matters: every indefinite integral represents a whole family of functions
Purpose: Build a clear understanding of integrals and antiderivatives so you can compute indefinite integrals using the main antiderivative rules (power rule, exponential, logarithmic, and basic trigonometric integrals), recognize common patterns for u-substitution (reverse chain rule), and evaluate definite integrals using the Fundamental Theorem of Calculus. You’ll also practice interpreting what an integral means: area, net area, and accumulation.
Success criteria
Explain the relationship between derivatives and antiderivatives: if \(F'(x)=f(x)\), then \(\int f(x)\,dx=F(x)+C\).
Use linearity of integrals: \(\int (af+bg)\,dx=a\int f\,dx+b\int g\,dx\).
Apply the power rule for integration and handle the special case \(n=-1\) correctly.
Compute common log and exponential integrals, including \(\int \frac{1}{x}\,dx=\ln|x|+C\) and \(\int e^x\,dx=e^x+C\).
Compute core trigonometric integrals, including \(\int \sec^2x\,dx=\tan x+C\) and \(\int \csc^2x\,dx=-\cot x+C\).
Recognize inverse trig patterns like \(\int \frac{1}{1+x^2}\,dx=\arctan(x)+C\).
Use simple u-substitution when you see “inside function + its derivative,” e.g. \(\int \frac{2x}{x^2+1}\,dx=\ln(x^2+1)+C\).
Evaluate definite integrals using an antiderivative: \(\int_a^b f(x)\,dx=F(b)-F(a)\).
Key vocabulary
Indefinite integral: \(\int f(x)\,dx\), a family of antiderivatives \(F(x)+C\).
Antiderivative: a function \(F\) such that \(F'(x)=f(x)\).
Constant of integration: \(+C\), needed because derivatives ignore constants.
Definite integral: \(\int_a^b f(x)\,dx\), the net accumulation / signed area from \(a\) to \(b\).
Fundamental Theorem of Calculus: if \(F'(x)=f(x)\), then \(\int_a^b f(x)\,dx=F(b)-F(a)\).
u-substitution: a method that reverses the chain rule by substituting \(u=g(x)\).
Quick pre-check
Pre-check 1: What is \(\displaystyle \int 3x^2\,dx\)?
Hint: Use \(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\).
Pre-check 2: What is \(\displaystyle \int \frac{1}{x}\,dx\)?
Hint: The derivative of \(\ln|x|\) is \(1/x\) for \(x≠ 0\).
Antiderivative Basics
Indefinite integrals, antiderivatives, and linearity
Learning goal: Understand what \(\int f(x)\,dx\) means and compute basic antiderivatives using linearity.
Key idea
An indefinite integral represents a family of antiderivatives: \[ \int f(x)\,dx = F(x) + C \quad \text{where } F'(x)=f(x). \] The constant of integration \(+C\) is required because differentiating any constant gives \(0\).
Use linearity: \[ \int (2x+3)\,dx = \int 2x\,dx + \int 3\,dx. \] Now integrate each piece: \[ \int 2x\,dx = x^2,\quad \int 3\,dx = 3x. \] So, \[ \int (2x+3)\,dx = x^2 + 3x + C. \]
Try it
Try it 1: What is \(\displaystyle \int 4x^3\,dx\)?
Hint: \(\int 4x^3\,dx = 4\cdot\dfrac{x^4}{4}+C\).
Try it 2: What is \(\displaystyle \int e^{0}\,dx\)?
Hint: \(e^0=1\), so you are integrating the constant \(1\).
Summary
\(\int f(x)\,dx = F(x)+C\) where \(F'(x)=f(x)\).
Use linearity to break integrals into simpler pieces.
Power Rule & Logs
The power rule for integration and the \(\int \frac{1}{x}\,dx\) exception
Learning goal: Apply the power rule correctly and remember the special logarithm case.
Key idea
The most-used rule for antiderivatives is the power rule: \[ \int x^n\,dx=\frac{x^{n+1}}{n+1}+C \quad (n≠ -1). \] The exponent increases by \(1\), then you divide by the new exponent.
But when \(n=-1\), the formula would divide by \(0\). That special case is: \[ \int \frac{1}{x}\,dx=\ln|x|+C. \]
Rewrite \(\frac{5}{x^2}=5x^{-2}\). Then apply the power rule with \(n=-2\): \[ \int 5x^{-2}\,dx = 5\cdot\frac{x^{-1}}{-1}+C = -\frac{5}{x}+C. \]
Try it
Try it 1: What is \(\displaystyle \int x^{1/2}\,dx\)?
Hint: Increase the exponent: \(1/2 \to 3/2\), then divide by \(3/2\).
Try it 2: What is \(\displaystyle \int \frac{3}{x}\,dx\)?
Hint: Pull out constants: \(\int \frac{3}{x}\,dx=3\int \frac{1}{x}\,dx\).
Summary
Power rule: \(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\) for \(n≠ -1\).
Special case: \(\int \dfrac{1}{x}\,dx=\ln|x|+C\).
Trig Integrals
Common trigonometric and inverse trigonometric integrals
Learning goal: Memorize the most common trig antiderivatives and recognize key inverse trig patterns.
Core trig integrals to know cold
\(\displaystyle \int \cos x\,dx=\sin x + C\)
\(\displaystyle \int \sin x\,dx=-\cos x + C\)
\(\displaystyle \int \sec^2 x\,dx=\tan x + C\)
\(\displaystyle \int \csc^2 x\,dx=-\cot x + C\)
\(\displaystyle \int \sec x\tan x\,dx=\sec x + C\)
\(\displaystyle \int \csc x\cot x\,dx=-\csc x + C\)
Inverse trig pattern
A must-know inverse trig antiderivative is: \[ \int \frac{1}{1+x^2}\,dx=\arctan(x)+C. \] This is the antiderivative because \(\dfrac{d}{dx}\arctan(x)=\dfrac{1}{1+x^2}\).
Know the standard trig antiderivatives and the sign patterns.
\(\int \dfrac{1}{1+x^2}\,dx=\arctan(x)+C\) is a key inverse trig result.
u-Substitution
u-substitution: the reverse chain rule for integrals
Learning goal: Spot common “inside function + derivative” patterns and integrate quickly.
Key idea
If you can identify a composition \(f(g(x))\) together with \(g'(x)\), then the integral often simplifies by substituting \(u=g(x)\). In practice, you’re matching a derivative pattern: \[ \int f(g(x))\,g'(x)\,dx = \int f(u)\,du. \] You do not need full algebraic substitution for every problem — many early integrals are “pattern matches.”
Let \(u=x^2+1\). Then \(du=2x\,dx\). The integral becomes: \[ \int \frac{2x}{x^2+1}\,dx=\int \frac{1}{u}\,du=\ln|u|+C. \] Substitute back \(u=x^2+1\) (always positive), so: \[ \int \frac{2x}{x^2+1}\,dx=\ln(x^2+1)+C. \]
Try it
Try it 1: What is \(\displaystyle \int \tan(x)\sec^2(x)\,dx\)?
Hint: Let \(u=\tan x\). Then \(du=\sec^2 x\,dx\).
Try it 2: What is \(\displaystyle \int (2x+1)^3\,dx\)?
Hint: Let \(u=2x+1\). Then \(du=2\,dx\) so \(dx=\frac{1}{2}du\).
Summary
Look for an inside function \(u=g(x)\) and (a constant multiple of) its derivative \(g'(x)\,dx\).
u-substitution is the reverse chain rule for integrals.
Definite Integrals
Definite integrals and the Fundamental Theorem of Calculus
Learning goal: Evaluate definite integrals correctly using antiderivatives and understand the “net area” idea.
Key idea
A definite integral measures net accumulation from \(a\) to \(b\): \[ \int_a^b f(x)\,dx. \] If \(F'(x)=f(x)\), then the Fundamental Theorem of Calculus says: \[ \int_a^b f(x)\,dx = F(b)-F(a). \] Notice: there is no \(+C\) in a definite integral because the constants cancel when you subtract.
An antiderivative of \(x\) is \(\frac{x^2}{2}\). Apply FTC: \[ \int_0^1 x\,dx = \left[\frac{x^2}{2}\right]_0^1=\frac{1^2}{2}-\frac{0^2}{2}=\frac{1}{2}. \]
Try it
Try it 1: Find \(\displaystyle \int_{0}^{\pi/2} \cos x\,dx\).
Hint: An antiderivative of \(\cos x\) is \(\sin x\). Evaluate \(\sin(\pi/2)-\sin(0)\).
Try it 2: Evaluate \(\displaystyle \int_{0}^{1} 3x^2\,dx\).
Hint: \(\int 3x^2\,dx=x^3\). Evaluate \(x^3\) from 0 to 1.
Summary
Definite integrals use evaluation: \(\int_a^b f(x)\,dx=F(b)-F(a)\).
No \(+C\) in definite integrals (constants cancel).
Mixed Practice
Mixed integration patterns: powers, rationals, and quick recognition
Learning goal: Build speed by recognizing which rule applies: power rule, log form, trig, or u-substitution.
Key idea
Most early integration problems are about picking the right pattern fast:
Power rule: \(x^n\) with \(n≠ -1\)
Log form: \(\dfrac{1}{x}\) or \(\dfrac{g'(x)}{g(x)}\)
u-substitution: reverse chain rule, especially \(\int \dfrac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C\).
Definite integrals: \(\int_a^b f(x)\,dx=F(b)-F(a)\) (no \(+C\)).
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the integration pattern you need.
Practice set
Integrals & Antiderivatives practice questions with instant score
Answer all 10 questions below, then get your final score and a mistake review at the end so you know exactly what to improve.
0/10answered
Question 1Not answered
What is the integral of \(2x e^{x^2}\,dx\)?
Correct answer: D. \(e^{x^2}+C\)
Explanation: Reverse chain rule: let \(u=x^2\), then \(du=2x\,dx\), so \(\int2x e^{x^2}dx=\int e^u du=e^u+C=e^{x^2}+C\).
Question 2Not answered
What is the integral of \(x\,dx\)?
Correct answer: D. \(\tfrac{x^2}{2} + C\)
Explanation: Power rule: \(\int x^n dx = x^{n+1}/(n+1)+C\); here \(n=1\), so \(\int x dx = x^2/2 + C\).
Question 3Not answered
What is the integral of \(x^2\,dx\)?
Correct answer: A. \(\tfrac{x^3}{3} + C\)
Explanation: Power rule: \(\int x^n dx = x^{n+1}/(n+1)+C\); here \(n=2\), so \(\int x^2 dx = x^3/3 + C\).
Question 4Not answered
What is the integral of \(e^x\,dx\)?
Correct answer: A. \(e^x + C\)
Explanation: The antiderivative of \(e^x\) is itself: \(\int e^x dx = e^x + C\).
