Integral of theta*sin(1)/4

The calculator will find the integral/antiderivative of $$$\frac{\theta \sin{\left(1 \right)}}{4}$$$, with steps shown.

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Find $$$\int \frac{\theta \sin{\left(1 \right)}}{4}\, d\theta$$$.

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 $$$c=\frac{\sin{\left(1 \right)}}{4}$$$ and $$$f{\left(\theta \right)} = \theta$$$:

$${\color{red}{\int{\frac{\theta \sin{\left(1 \right)}}{4} d \theta}}} = {\color{red}{\left(\frac{\sin{\left(1 \right)} \int{\theta d \theta}}{4}\right)}}$$

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

$$\frac{\sin{\left(1 \right)} {\color{red}{\int{\theta d \theta}}}}{4}=\frac{\sin{\left(1 \right)} {\color{red}{\frac{\theta^{1 + 1}}{1 + 1}}}}{4}=\frac{\sin{\left(1 \right)} {\color{red}{\left(\frac{\theta^{2}}{2}\right)}}}{4}$$

Therefore,

$$\int{\frac{\theta \sin{\left(1 \right)}}{4} d \theta} = \frac{\theta^{2} \sin{\left(1 \right)}}{8}$$

Add the constant of integration:

$$\int{\frac{\theta \sin{\left(1 \right)}}{4} d \theta} = \frac{\theta^{2} \sin{\left(1 \right)}}{8}+C$$

Answer

$$$\int \frac{\theta \sin{\left(1 \right)}}{4}\, d\theta = \frac{\theta^{2} \sin{\left(1 \right)}}{8} + C$$$A

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Integral of $$$\frac{\theta \sin{\left(1 \right)}}{4}$$$

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