Integral of sin(2)^2/x

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

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

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 $$$c=\sin^{2}{\left(2 \right)}$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:

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

The integral of $$$\frac{1}{x}$$$ is $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

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

Therefore,

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

Add the constant of integration:

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

Answer

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

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

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