Integral of $$$x^{2} - 38 \sin{\left(x \right)}$$$

Find $$$\int \left(x^{2} - 38 \sin{\left(x \right)}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(x^{2} - 38 \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{x^{2} d x} - \int{38 \sin{\left(x \right)} d x}\right)}}$$

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

$$- \int{38 \sin{\left(x \right)} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{38 \sin{\left(x \right)} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{38 \sin{\left(x \right)} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=38$$$ and $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

$$\frac{x^{3}}{3} - {\color{red}{\int{38 \sin{\left(x \right)} d x}}} = \frac{x^{3}}{3} - {\color{red}{\left(38 \int{\sin{\left(x \right)} d x}\right)}}$$

The integral of the sine is $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$\frac{x^{3}}{3} - 38 {\color{red}{\int{\sin{\left(x \right)} d x}}} = \frac{x^{3}}{3} - 38 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Therefore,

$$\int{\left(x^{2} - 38 \sin{\left(x \right)}\right)d x} = \frac{x^{3}}{3} + 38 \cos{\left(x \right)}$$

Add the constant of integration:

$$\int{\left(x^{2} - 38 \sin{\left(x \right)}\right)d x} = \frac{x^{3}}{3} + 38 \cos{\left(x \right)}+C$$

$$$\int \left(x^{2} - 38 \sin{\left(x \right)}\right)\, dx = \left(\frac{x^{3}}{3} + 38 \cos{\left(x \right)}\right) + C$$$A