Integral of $$$\sin^{3}{\left(x \right)}$$$

The calculator will find the integral/antiderivative of $$$\sin^{3}{\left(x \right)}$$$, with steps shown.

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Find $$$\int \sin^{3}{\left(x \right)}\, dx$$$.


Strip out one sine and write everything else in terms of the cosine, using the formula $$$\sin^2\left(\alpha \right)=-\cos^2\left(\alpha \right)+1$$$ with $$$\alpha=x$$$:

$$\color{red}{\int{\sin^{3}{\left(x \right)} d x}} = \color{red}{\int{\left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} d x}}$$

Let $$$u=\cos{\left(x \right)}$$$.

Then $$$du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx$$$ (steps can be seen here), and we have that $$$\sin{\left(x \right)} dx = - du$$$.


$$\color{red}{\int{\left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} d x}} = \color{red}{\int{\left(u^{2} - 1\right)d u}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = 1 - u^{2}$$$:

$$\color{red}{\int{\left(u^{2} - 1\right)d u}} = \color{red}{\left(- \int{\left(1 - u^{2}\right)d u}\right)}$$

Integrate term by term:

$$- \color{red}{\int{\left(1 - u^{2}\right)d u}} = - \color{red}{\left(\int{1 d u} - \int{u^{2} d u}\right)}$$

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=1$$$:

$$\int{u^{2} d u} - \color{red}{\int{1 d u}} = \int{u^{2} d u} - \color{red}{u}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- u + \color{red}{\int{u^{2} d u}}=- u + \color{red}{\frac{u^{1 + 2}}{1 + 2}}=- u + \color{red}{\left(\frac{u^{3}}{3}\right)}$$

Recall that $$$u=\cos{\left(x \right)}$$$:

$$- \color{red}{u} + \frac{\color{red}{u}^{3}}{3} = - \color{red}{\cos{\left(x \right)}} + \frac{\color{red}{\cos{\left(x \right)}}^{3}}{3}$$


$$\int{\sin^{3}{\left(x \right)} d x} = \frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$

Add the constant of integration:

$$\int{\sin^{3}{\left(x \right)} d x} = \frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}+C$$

Answer: $$$\int{\sin^{3}{\left(x \right)} d x}=\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}+C$$$