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# 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

x3 dx
Solution

dx
Solution

### Example

(6x2 - 2x + 5) dx
Solution

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

cos2 x2 dx
Solution

## Integral of sec2 x

### Formula

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

cos 2xcos2 x dx
Solution

ex dx = ex + C
Derivative of ex

e2x - 1ex - 1 dx
Solution

## Integral of ax

### Formula

ax dx = 1ln aax + C

2x(3x - 1) dx
Solution

## Integral of1x

### Formula

1x dx = ln |x| + C

x2 - x + 1x dx
Solution

(2x - 1)8 dx
Solution

sin5 x cos x dx
Solution

## Integral of tan x

### Formula

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

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

(2x - 1)8 dx
Solution

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

dxa2 - x2
Solution

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.)

xex dx
Solution

x cos x dx
Solution

x2 sin x dx
Solution

ex sin x dx
Solution

## Integral of ln x

### Formula

ln x dx = x ln x - x + C

ln x dx
Solution

(ln x)2 dx
Solution