Calculus Practice

Integrals Practice

Practice antiderivatives, indefinite and definite integrals, the Fundamental Theorem of Calculus, \(u\)-substitution, trig integrals, and integral properties.

Start Practicing Review Integrals

Try each problem first, then click Show solution to check your work and see the complete step-by-step solution.

Keep These Rules Nearby

Integrals Essentials

Power Rule

\[ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \]

\(n\neq -1\)

Add one to the exponent, then divide by the new exponent.

Fundamental Theorem

\[ \int_a^b f(x)\,dx = F(b)-F(a) \]

Find an antiderivative, then evaluate upper minus lower.

\(u\)-Substitution

\[ u=g(x) \] \[ du=g'(x)\,dx \]

Use substitution to reverse the chain rule.

Download Practice PDF Review Integrals

Level 1

Antiderivatives and Basic Rules

Start with simple antiderivatives, constants, and the integration power rule.

Problem 1

Antiderivative

Find the indefinite integral: \[ \int 6x\,dx \]

Step-by-Step Solution

We need a function whose derivative is \(6x\).

\[ \int 6x\,dx = 6 \int x\,dx \] \[ = 6 \left( \frac{x^2}{2} \right) + C \] \[ = 3x^2+C \]
Answer: \[ \boxed{ 3x^2+C } \]

Problem 2

Constant Rule

Evaluate: \[ \int 7\,dx \]

Step-by-Step Solution

The antiderivative of a constant \(c\) is

\[ cx+C. \]
\[ \int 7\,dx = 7x+C \]
Answer: \[ \boxed{ 7x+C } \]

Problem 3

Power Rule

Evaluate: \[ \int x^5\,dx \]

Step-by-Step Solution

Add one to the exponent:

\[ 5+1=6. \]

Divide by the new exponent:

\[ \int x^5\,dx = \frac{x^6}{6} + C \]
Answer: \[ \boxed{ \frac{x^6}{6}+C } \]

Level 2

Powers, Polynomials, and Special Integrals

Work with negative and fractional powers, polynomial integrals, and the special reciprocal rule.

Problem 4

Polynomial

Evaluate: \[ \int \left( 4x^3-6x^2+5x-8 \right) dx \]

Step-by-Step Solution

Integrate each term separately.

\[ \int 4x^3\,dx = x^4 \] \[ \int -6x^2\,dx = -2x^3 \] \[ \int 5x\,dx = \frac52x^2 \] \[ \int -8\,dx = -8x \]
Answer: \[ \boxed{ x^4 - 2x^3 + \frac52x^2 - 8x + C } \]

Problem 5

Negative Exponent

Evaluate: \[ \int x^{-3}\,dx \]

Step-by-Step Solution

Apply the power rule.

\[ \int x^{-3}\,dx = \frac{ x^{-2} }{ -2 } + C \] \[ = -\frac12x^{-2}+C \]

Rewrite using positive exponents:

\[ -\frac12x^{-2} = -\frac{1}{2x^2}. \]
Answer: \[ \boxed{ -\frac{1}{2x^2}+C } \]

Problem 6

Radical

Evaluate: \[ \int \sqrt{x}\,dx \]

Step-by-Step Solution

Rewrite the radical:

\[ \sqrt{x} = x^{1/2}. \]

Apply the power rule.

\[ \int x^{1/2}\,dx = \frac{ x^{3/2} }{ 3/2 } + C \] \[ = \frac23 x^{3/2} + C \]
Answer: \[ \boxed{ \frac23x^{3/2}+C } \]

Problem 7

Reciprocal Rule

Evaluate: \[ \int \frac{1}{x}\,dx \]

Step-by-Step Solution

The power rule does not apply when the exponent is \(-1\).

Use the special reciprocal rule:

\[ \int \frac{1}{x}\,dx = \ln|x|+C \]
Answer: \[ \boxed{ \ln|x|+C } \]

Level 3

Definite Integrals and the Fundamental Theorem

Evaluate definite integrals and connect integration with differentiation.

Problem 8

Definite Integral

Evaluate: \[ \int_1^3 2x\,dx \]

Step-by-Step Solution

An antiderivative of \(2x\) is

\[ x^2. \]

Evaluate upper minus lower:

\[ \left[ x^2 \right]_1^3 = 3^2-1^2 \] \[ = 9-1 = 8 \]
Answer: \[ \boxed{8} \]

Problem 9

Fundamental Theorem

Differentiate: \[ G(x) = \int_2^x \left( t^3+1 \right) dt \]

Step-by-Step Solution

By the Fundamental Theorem of Calculus,

\[ \frac{d}{dx} \left[ \int_a^x f(t)\,dt \right] = f(x). \]

Replace \(t\) by \(x\) in the integrand.

Answer: \[ \boxed{ G'(x) = x^3+1 } \]

Problem 10

FTC + Chain Rule

Differentiate: \[ G(x) = \int_0^{x^2} \cos t\,dt \]

Step-by-Step Solution

Evaluate the integrand at the upper bound:

\[ \cos(x^2). \]

Then multiply by the derivative of \(x^2\).

\[ \frac{d}{dx}(x^2) = 2x. \]
\[ G'(x) = \cos(x^2)(2x) \] \[ = 2x\cos(x^2) \]
Answer: \[ \boxed{ G'(x) = 2x\cos(x^2) } \]

Level 4

Trigonometric Integrals and \(u\)-Substitution

Apply basic trigonometric antiderivatives and reverse the chain rule with substitution.

Problem 11

Trig Integral

Evaluate: \[ \int \left( 3\cos x - 2\sin x \right) dx \]

Step-by-Step Solution

Recall:

\[ \int \cos x\,dx = \sin x \] \[ \int \sin x\,dx = -\cos x. \]
\[ \int 3\cos x\,dx = 3\sin x \] \[ \int -2\sin x\,dx = 2\cos x \]
Answer: \[ \boxed{ 3\sin x + 2\cos x + C } \]

Problem 12

\(u\)-Substitution

Evaluate: \[ \int 2x(x^2+1)^4\,dx \]

Step-by-Step Solution

Let

\[ u=x^2+1. \]

Then

\[ du=2x\,dx. \]

Rewrite:

\[ \int u^4\,du \] \[ = \frac{u^5}{5} + C \]

Substitute back:

Answer: \[ \boxed{ \frac{ (x^2+1)^5 }{ 5 } + C } \]

Problem 13

Constant Factor

Evaluate: \[ \int x(x^2+4)^3\,dx \]

Step-by-Step Solution

Let

\[ u=x^2+4. \]

Then

\[ du=2x\,dx. \]

Therefore,

\[ x\,dx = \frac12du. \]
\[ \frac12 \int u^3\,du \] \[ = \frac12 \cdot \frac{u^4}{4} + C \] \[ = \frac{u^4}{8} + C \]
Answer: \[ \boxed{ \frac{ (x^2+4)^4 }{ 8 } + C } \]

Problem 14

Definite \(u\)-Substitution

Evaluate: \[ \int_0^1 2x(x^2+1)^2\,dx \]

Step-by-Step Solution

Let

\[ u=x^2+1, \qquad du=2x\,dx. \]

Convert the bounds:

\[ x=0 \Rightarrow u=1 \] \[ x=1 \Rightarrow u=2. \]

Rewrite:

\[ \int_1^2u^2\,du \] \[ = \left[ \frac{u^3}{3} \right]_1^2 \] \[ = \frac83-\frac13 \] \[ = \frac73 \]
Answer: \[ \boxed{ \frac73 } \]

Level 5

Integral Properties and Mixed Review

Finish with integral properties, symmetry, signed area, and mixed concepts.

Problem 15

Integral Properties

Suppose \[ \int_1^4 f(x)\,dx=7. \] Find \[ \int_4^1 f(x)\,dx. \]

\(7\)
\(-7\)
\(0\)

Step-by-Step Solution

Reversing the bounds changes the sign of a definite integral.

\[ \int_4^1f(x)\,dx = - \int_1^4f(x)\,dx \] \[ = -7 \]
Answer: \[ \boxed{-7} \]

Problem 16

Symmetry

Suppose \(f\) is an odd function. Evaluate \[ \int_{-5}^{5} f(x)\,dx. \]

\(0\)
\(5\)
Cannot be determined

Step-by-Step Solution

For an odd function,

\[ f(-x)=-f(x). \]

Over an interval symmetric about zero, the positive and negative signed areas cancel.

\[ \int_{-a}^{a} f(x)\,dx = 0 \]
Answer: \[ \boxed{0} \]

Before You Finish

Integrals Checklist

  1. Recognize an antiderivative as a function whose derivative equals the integrand.
  2. Include \(+C\) when evaluating indefinite integrals.
  3. Apply the integration power rule correctly.
  4. Rewrite radicals and negative powers when useful.
  5. Integrate polynomials term by term.
  6. Remember \(\int \frac1x\,dx=\ln|x|+C\).
  7. Use the Fundamental Theorem to evaluate definite integrals.
  8. Evaluate definite integrals as upper minus lower.
  9. Differentiate accumulation functions using the Fundamental Theorem.
  10. Apply the chain rule when the upper limit is a function of \(x\).
  11. Know the basic trig antiderivatives.
  12. Use \(u\)-substitution to reverse the chain rule.
  13. Adjust correctly for missing constant factors.
  14. Convert bounds or substitute back in definite \(u\)-substitution.
  15. Reverse bounds by changing the sign.
  16. Use even and odd symmetry when possible.

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