Integrals & Antiderivatives Practice Questions, Quiz, and Step-by-Step Lesson - improve your math skills with focused questions and clear explanations.
Integrals & Antiderivatives Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page 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.
How this integrals and antiderivatives practice works
1. Take the quiz: answer the integrals and antiderivatives questions at the top of the page.
2. Open the lesson (optional): review antiderivative rules, trig/exponential/log integrals, u-substitution patterns, and definite integrals.
3. Retry: return to the quiz 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
u-substitution: spot an "inside function" and its derivative (reverse chain rule)
Patterns like \(\int \dfrac{2x}{x^2+1}\,dx=\ln(x^2+1)+C\)
Definite integrals: compute \(\int_a^b f(x)\,dx=F(b)-F(a)\) using the Fundamental Theorem of Calculus
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing integrals and antiderivatives.
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Integrals & Antiderivatives
Step-by-step guide
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Integrals & Antiderivatives Lesson
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Lesson Overview
Lesson overview
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.
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