# Integral calculus Calculator

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### Difficult Problems

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

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