1
Here, we show you a step-by-step solved example of trigonometric integrals. This solution was automatically generated by our smart calculator:
$\int\sin\left(x\right)^4dx$
Intermediate steps
Apply the formula: $\int\sin\left(\theta \right)^ndx$$=\frac{-\sin\left(\theta \right)^{\left(n-1\right)}\cos\left(\theta \right)}{n}+\frac{n-1}{n}\int\sin\left(\theta \right)^{\left(n-2\right)}dx$, where $n=4$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}+\frac{4-1}{4}\int\sin\left(x\right)^{2}dx$
Add the values $4$ and $-1$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}+\frac{3}{4}\int\sin\left(x\right)^{2}dx$
2
Apply the formula: $\int\sin\left(\theta \right)^ndx$$=\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 $n=4$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}+\frac{3}{4}\int\sin\left(x\right)^{2}dx$
Explain this step further
3
Multiply the single term $\frac{3}{4}$ by each term of the polynomial $\left(\frac{1}{2}x-\frac{1}{4}\sin\left(2x\right)\right)$
$\frac{1}{2}\cdot \frac{3}{4}x-\frac{1}{4}\cdot \frac{3}{4}\sin\left(2x\right)$
Intermediate steps
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: $\frac{1}{2}\cdot \frac{3}{4}x-\frac{1}{4}\cdot \frac{3}{4}\sin\left(2x\right)$
$\frac{1}{2}\cdot \frac{3}{4}x-\frac{1}{4}\cdot \frac{3}{4}\sin\left(2x\right)$
Explain this step further
5
Gather the results of all integrals
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{1}{4}\cdot \frac{3}{4}\sin\left(2x\right)+\frac{1}{2}\cdot \frac{3}{4}x$
Intermediate steps
Multiplying fractions $-\frac{1}{4} \times \frac{3}{4}$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}+\frac{-3}{4\cdot 4}\sin\left(2x\right)+\frac{1}{2}\cdot \frac{3}{4}x$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{1}{2}\cdot \frac{3}{4}x$
6
Multiplying fractions $-\frac{1}{4} \times \frac{3}{4}$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{1}{2}\cdot \frac{3}{4}x$
Explain this step further
Intermediate steps
Multiplying fractions $\frac{1}{2} \times \frac{3}{4}$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{1\cdot 3}{2\cdot 4}x$
Multiplying fractions $-\frac{1}{4} \times \frac{3}{4}$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}+\frac{-3}{4\cdot 4}\sin\left(2x\right)+\frac{1}{2}\cdot \frac{3}{4}x$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{1}{2}\cdot \frac{3}{4}x$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{1\cdot 3}{8}x$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{3}{8}x$
7
Multiplying fractions $\frac{1}{2} \times \frac{3}{4}$
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{3}{8}x$
Explain this step further
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
$\frac{-\sin\left(x\right)^{3}\cos\left(x\right)}{4}-\frac{3}{16}\sin\left(2x\right)+\frac{3}{8}x+C_0$