Ximpledu

Indefinite Integral

See how to solve an indefinite integral.
25 examples and their solutions.

Integral: Definition

Definition

Integral

Derivative
Integral and derivative are the opposite operations.
f'(x) is the derivative of f(x).
So f(x) is the integral of f'(x).
This is why integral is also called antiderivative.
Integral

Derivative
So, finding ∫ f(x) dx is finding the function whose derivative is f(x).
∫ f(x) dx is read as [integral f x d x].

Integral of a⋅f(x) + b⋅g(x)

Property

[af(x) + bg(x)] dx
= af(x) dx + bg(x) dx

Integral of xn

Formula

xn dx = 1n + 1xn + 1 + C
xn
Write the reciprocal of (n + 1): 1/(n + 1).
Write xn + 1.
Don't forget to write the constant term +C.

The derivative of [1/(n + 1)]xn + 1 + C is xn.
→ Integral of xn is [1/(n + 1)]xn + 1 + C.
Derivative of xn

Example

x3 dx
Solution

Example

dx
Solution

Example

(6x2 - 2x + 5) dx
Solution

Example

x dx
Solution

Integral of sin x

Formula

sin x dx = -cos x + C
Derivative of cos x

Example

(sin x2 + cos x2)2 dx
Solution

Integral of cos x

Formula

cos x dx = sin x + C
Derivative of sin x

Example

cos2 x2 dx
Solution

Integral of sec2 x

Formula

sec2 x dx = tan x + C
Derivative of tan x

Example

cos 2xcos2 x dx
Solution

Integral of ex

Formula

ex dx = ex + C
Derivative of ex

Example

e2x - 1ex - 1 dx
Solution

Integral of ax

Formula

ax dx = 1ln aax + C

Example

2x(3x - 1) dx
Solution

Integral of 1x

Formula

1x dx = ln |x| + C

Example

x2 - x + 1x dx
Solution

Integral by Substitution

Example

(2x - 1)8 dx
Solution

Example

sin5 x cos x dx
Solution

Integral of tan x

Formula

tan x dx = -ln |cos x| + C

Example

tan x dx
Solution

Integral of f(ax + b)

Formula

f(ax + b) dx = 1aF(ax + b) + C
F(x): Integral of f(x)
1/a: Reciprocal of a

Example

(2x - 1)8 dx
Solution

Example

1x - 3 dx
Solution

Integral of a Rational Expression

Example

x2 - 3x + 5x - 1 dx
Solution

Example

3x - 2x(x - 1) dx
Solution

Trigonometric Substitution

Formula

a2 - x2 x = a sin θ
a2 + x2 x = a tan θ
Special case of integral by substitution

Example

dxa2 - x2
Solution

Example

dxa2 + x2
Solution

Integral by Parts

Formula

uv' dx = uv - u'v dx
Proof

How to Determine u and v'

ln x, loga x
x, x2, x3, ...
sin x, cos x
ex, ax
uv'
Upper function → u
(= The integral is complex.)
Lower function → v'
(= The integral is simple.)

Example

xex dx
Solution

Example

x cos x dx
Solution

Example

x2 sin x dx
Solution

Example

ex sin x dx
Solution

Integral of ln x

Formula

ln x dx = x ln x - x + C

Example

ln x dx
Solution

Example

(ln x)2 dx
Solution