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Trigonometric Integrals Calculator

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1

Here, we show you a step-by-step solved example of trigonometric integrals. This solution was automatically generated by our smart calculator:

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

Apply the formula: $\int\sin\left(\theta \right)^ndx$$=\frac{-\sin\left(\theta \right)^{\left(n-1\right)}\cos\left(\theta \right)}{n}+\frac{n-1}{n}\int\sin\left(\theta \right)^{\left(n-2\right)}dx$, where $n=4$

$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}+\frac{3}{4}\int\sin\left(x\right)^{2}dx$
3

Multiply the single term $\frac{3}{4}$ by each term of the polynomial $\left(\frac{1}{2}x-\frac{1}{4}\sin\left(2x\right)\right)$

$\frac{1}{2}\cdot \frac{3}{4}x-\frac{1}{4}\cdot \frac{3}{4}\sin\left(2x\right)$

Apply the formula: $\int\sin\left(\theta \right)^2dx$$=\frac{1}{2}\theta -\frac{1}{4}\sin\left(2\theta \right)+C$

$\frac{3}{4}\left(\frac{1}{2}x-\frac{1}{4}\sin\left(2x\right)\right)$
4

The integral $\frac{3}{4}\int\sin\left(x\right)^{2}dx$ results in: $\frac{1}{2}\cdot \frac{3}{4}x-\frac{1}{4}\cdot \frac{3}{4}\sin\left(2x\right)$

$\frac{1}{2}\cdot \frac{3}{4}x-\frac{1}{4}\cdot \frac{3}{4}\sin\left(2x\right)$
5

Gather the results of all integrals

$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{1}{4}\cdot \frac{3}{4}\sin\left(2x\right)+\frac{1}{2}\cdot \frac{3}{4}x$
6

Multiplying fractions $-\frac{1}{4} \times \frac{3}{4}$

$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{1}{2}\cdot \frac{3}{4}x$
7

Multiplying fractions $\frac{1}{2} \times \frac{3}{4}$

$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{3}{8}x$
8

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{3}{8}x+C_0$

Final answer to the problem

$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{3}{8}x+C_0$

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