Integral of x*sin(2)

The calculator will find the integral/antiderivative of $$$x \sin{\left(2 \right)}$$$, with steps shown.

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

The trigonometric functions expect the argument in radians. To enter the argument in degrees, multiply it by pi/180, e.g. write 45° as 45*pi/180, or use the appropriate function adding 'd', e.g. write sin(45°) as sind(45).

Solution

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\sin{\left(2 \right)}$$$ and $$$f{\left(x \right)} = x$$$:

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

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

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

Therefore,

$$\int{x \sin{\left(2 \right)} d x} = \frac{x^{2} \sin{\left(2 \right)}}{2}$$

Add the constant of integration:

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

Answer

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

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

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