Integral of e^x*tan(e)^x

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

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Your Input

Find $$$\int e^{x} \tan^{x}{\left(e \right)}\, dx$$$.

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

The input is rewritten: $$$\int{e^{x} \tan^{x}{\left(e \right)} d x}=\int{\left(e \tan{\left(e \right)}\right)^{x} d x}$$$.

Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=e \tan{\left(e \right)}$$$:

$${\color{red}{\int{\left(e \tan{\left(e \right)}\right)^{x} d x}}} = {\color{red}{\frac{\left(e \tan{\left(e \right)}\right)^{x}}{\ln{\left(e \tan{\left(e \right)} \right)}}}}$$

Therefore,

$$\int{\left(e \tan{\left(e \right)}\right)^{x} d x} = \frac{\left(e \tan{\left(e \right)}\right)^{x}}{\ln{\left(e \tan{\left(e \right)} \right)}}$$

Simplify:

$$\int{\left(e \tan{\left(e \right)}\right)^{x} d x} = \frac{e^{x} \tan^{x}{\left(e \right)}}{\ln{\left(- \tan{\left(e \right)} \right)} + 1 + i \pi}$$

Add the constant of integration:

$$\int{\left(e \tan{\left(e \right)}\right)^{x} d x} = \frac{e^{x} \tan^{x}{\left(e \right)}}{\ln{\left(- \tan{\left(e \right)} \right)} + 1 + i \pi}+C$$

Answer

$$$\int e^{x} \tan^{x}{\left(e \right)}\, dx = \frac{e^{x} \tan^{x}{\left(e \right)}}{\ln\left(- \tan{\left(e \right)}\right) + 1 + i \pi} + C$$$A

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Integral of $$$e^{x} \tan^{x}{\left(e \right)}$$$

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