Practice Integral in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

The reverse operation of differentiation; it also computes the exact area under a curve between two points.

If derivative gives rate, integral gives total. Derivative of position = velocity; integral of velocity = position.

Showing a random 20 of 50 problems.

Example 1

medium
Find โˆซsinโกxโ€‰dx\int \sin x \, dx.

Example 2

easy
Find โˆซ(4x3+6x)โ€‰dx\int (4x^3 + 6x) \, dx

Example 3

easy
Find โˆซx5โ€‰dx\int x^5 \, dx.

Example 4

easy
Evaluate โˆซ032xโ€‰dx\int_0^3 2x \, dx.

Example 5

medium
Find โˆซ(x2โˆ’4x+5)โ€‰dx\int (x^2 - 4x + 5) \, dx.

Example 6

medium
Find โˆซ1x3โ€‰dx\int \dfrac{1}{x^3} \, dx.

Example 7

easy
Find โˆซ3โ€‰dx\int 3 \, dx.

Example 8

challenge
Find โˆซ(x+1)2โ€‰dx\int (x+1)^2 \, dx by expanding first.

Example 9

easy
Find โˆซx2โ€‰dx\int x^2 \, dx.

Example 10

easy
Find โˆซx3โ€‰dx\int x^3 \, dx.

Example 11

easy
Find โˆซ4x3โ€‰dx\int 4x^3 \, dx.

Example 12

medium
Find โˆซ(x3+3x2โˆ’2x+1)โ€‰dx\int (x^3 + 3x^2 - 2x + 1) \, dx.

Example 13

easy
Find โˆซ(5x2โˆ’3x+7)โ€‰dx\int (5x^2 - 3x + 7) \, dx

Example 14

medium
Find โˆซ(2ex+3cosโกx)โ€‰dx\int (2e^x + 3\cos x) \, dx.

Example 15

hard
A particle has velocity v(t)=3t2โˆ’6tv(t) = 3t^2 - 6t m/s. Find its displacement from t=0t=0 to t=4t=4.

Example 16

hard
Find โˆซxex2โ€‰dx\int xe^{x^2} \, dx.

Example 17

easy
Find โˆซ(2x+1)โ€‰dx\int (2x + 1) \, dx.

Example 18

hard
Evaluate โˆซ01(3x2+2x+1)โ€‰dx\int_0^1 (3x^2 + 2x + 1) \, dx.

Example 19

easy
Find โˆซcosโกxโ€‰dx\int \cos x \, dx.

Example 20

medium
Evaluate โˆซโˆ’11(x3+x)โ€‰dx\int_{-1}^{1} (x^3 + x) \, dx.