Integral of cos(2)*tanh(eta)

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

Related calculator: Definite and Improper Integral Calculator

Your Input

Find $$$\int \cos{\left(2 \right)} \tanh{\left(\eta \right)}\, d\eta$$$.

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 $$$\int c f{\left(\eta \right)}\, d\eta = c \int f{\left(\eta \right)}\, d\eta$$$ with $$$c=\cos{\left(2 \right)}$$$ and $$$f{\left(\eta \right)} = \tanh{\left(\eta \right)}$$$:

$${\color{red}{\int{\cos{\left(2 \right)} \tanh{\left(\eta \right)} d \eta}}} = {\color{red}{\cos{\left(2 \right)} \int{\tanh{\left(\eta \right)} d \eta}}}$$

Rewrite the hyperbolic tangent as $$$\tanh\left(\eta\right)=\frac{\sinh\left(\eta\right)}{\cosh\left(\eta\right)}$$$:

$$\cos{\left(2 \right)} {\color{red}{\int{\tanh{\left(\eta \right)} d \eta}}} = \cos{\left(2 \right)} {\color{red}{\int{\frac{\sinh{\left(\eta \right)}}{\cosh{\left(\eta \right)}} d \eta}}}$$

Let $$$u=\cosh{\left(\eta \right)}$$$.

Then $$$du=\left(\cosh{\left(\eta \right)}\right)^{\prime }d\eta = \sinh{\left(\eta \right)} d\eta$$$ (steps can be seen »), and we have that $$$\sinh{\left(\eta \right)} d\eta = du$$$.

Thus,

$$\cos{\left(2 \right)} {\color{red}{\int{\frac{\sinh{\left(\eta \right)}}{\cosh{\left(\eta \right)}} d \eta}}} = \cos{\left(2 \right)} {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$\cos{\left(2 \right)} {\color{red}{\int{\frac{1}{u} d u}}} = \cos{\left(2 \right)} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Recall that $$$u=\cosh{\left(\eta \right)}$$$:

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} \cos{\left(2 \right)} = \ln{\left(\left|{{\color{red}{\cosh{\left(\eta \right)}}}}\right| \right)} \cos{\left(2 \right)}$$

Therefore,

$$\int{\cos{\left(2 \right)} \tanh{\left(\eta \right)} d \eta} = \ln{\left(\cosh{\left(\eta \right)} \right)} \cos{\left(2 \right)}$$

Add the constant of integration:

$$\int{\cos{\left(2 \right)} \tanh{\left(\eta \right)} d \eta} = \ln{\left(\cosh{\left(\eta \right)} \right)} \cos{\left(2 \right)}+C$$

Answer

$$$\int \cos{\left(2 \right)} \tanh{\left(\eta \right)}\, d\eta = \ln\left(\cosh{\left(\eta \right)}\right) \cos{\left(2 \right)} + C$$$A

Integral of $$$\cos{\left(2 \right)} \tanh{\left(\eta \right)}$$$

Your Input

Solution

Answer