Integral of theta*sin(2)

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

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Find $$$\int \theta \sin{\left(2 \right)}\, d\theta$$$.

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 $$$\int c f{\left(\theta \right)}\, d\theta = c \int f{\left(\theta \right)}\, d\theta$$$ with $$$c=\sin{\left(2 \right)}$$$ and $$$f{\left(\theta \right)} = \theta$$$:

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

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

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

Therefore,

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

Add the constant of integration:

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

Answer

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

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

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