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Integral calculus Calculator

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1

Example

$\int\sin\left(x\right)^4dx$
2

Applying a sine identity in order to reduce the exponent: $\displaystyle\sin(\theta)=\sqrt{\frac{1-\cos(2\theta)}{2}}$

$\int\left(\frac{1-\cos\left(2x\right)}{2}\right)^{2}dx$
3

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\int\frac{\left(1-\cos\left(2x\right)\right)^{2}}{4}dx$
4

Take the constant out of the integral

$\frac{1}{4}\int\left(1-\cos\left(2x\right)\right)^{2}dx$
5

Expanding the polynomial

$\frac{1}{4}\int\left(\cos\left(2x\right)^2-2\cos\left(2x\right)+1\right)dx$
6

The integral of a sum of two or more functions is equal to the sum of their integrals

$\frac{1}{4}\left(\int\cos\left(2x\right)^2dx+\int-2\cos\left(2x\right)dx+\int1dx\right)$
7

The integral of a constant is equal to the constant times the integral's variable

$\frac{1}{4}\left(\int\cos\left(2x\right)^2dx+\int-2\cos\left(2x\right)dx+x\right)$
8

Take the constant out of the integral

$\frac{1}{4}\left(\int\cos\left(2x\right)^2dx-2\int\cos\left(2x\right)dx+x\right)$
9

Apply the formula: $\int\cos\left(x\cdot a\right)dx$$=\frac{1}{a}\sin\left(x\cdot a\right)$, where $a=2$

$\frac{1}{4}\left(\int\cos\left(2x\right)^2dx-2\cdot \frac{1}{2}\sin\left(2x\right)+x\right)$
10

Multiply $\frac{1}{2}$ times $-2$

$\frac{1}{4}\left(\int\cos\left(2x\right)^2dx-\sin\left(2x\right)+x\right)$
11

Multiplying polynomials

$\frac{1}{4}\int\cos\left(2x\right)^2dx-\frac{1}{4}\sin\left(2x\right)+\frac{1}{4}x$
12

Apply the formula: $\sin\left(2x\right)$$=2\cos\left(x\right)\sin\left(x\right)$

$\frac{1}{4}\int\cos\left(2x\right)^2dx-\frac{1}{4}\cdot 2\cos\left(x\right)\sin\left(x\right)+\frac{1}{4}x$
13

Multiply $2$ times $-\frac{1}{4}$

$\frac{1}{4}\int\cos\left(2x\right)^2dx-\frac{1}{2}\cos\left(x\right)\sin\left(x\right)+\frac{1}{4}x$
14

Apply the formula: $\cos\left(x\right)^2$$=\frac{\cos\left(2x\right)+1}{2}$, where $x=2x$

$\frac{1}{4}\int\frac{\cos\left(4x\right)+1}{2}dx-\frac{1}{2}\cos\left(x\right)\sin\left(x\right)+\frac{1}{4}x$
15

Take the constant out of the integral

$\frac{1}{4}\cdot \frac{1}{2}\int\left(\cos\left(4x\right)+1\right)dx-\frac{1}{2}\cos\left(x\right)\sin\left(x\right)+\frac{1}{4}x$
16

Multiply $\frac{1}{2}$ times $\frac{1}{4}$

$\frac{1}{8}\int\left(\cos\left(4x\right)+1\right)dx-\frac{1}{2}\cos\left(x\right)\sin\left(x\right)+\frac{1}{4}x$
17

The integral of a sum of two or more functions is equal to the sum of their integrals

$\frac{1}{8}\left(\int\cos\left(4x\right)dx+\int1dx\right)-\frac{1}{2}\cos\left(x\right)\sin\left(x\right)+\frac{1}{4}x$
18

The integral of a constant is equal to the constant times the integral's variable

$\frac{1}{8}\left(\int\cos\left(4x\right)dx+x\right)-\frac{1}{2}\cos\left(x\right)\sin\left(x\right)+\frac{1}{4}x$
19

Apply the formula: $\int\cos\left(x\cdot a\right)dx$$=\frac{1}{a}\sin\left(x\cdot a\right)$, where $a=4$

$\frac{1}{8}\left(\frac{1}{4}\sin\left(4x\right)+x\right)-\frac{1}{2}\cos\left(x\right)\sin\left(x\right)+\frac{1}{4}x$
20

Add the constant of integration

$\frac{1}{8}\left(\frac{1}{4}\sin\left(4x\right)+x\right)-\frac{1}{2}\cos\left(x\right)\sin\left(x\right)+\frac{1}{4}x+C_0$