What is the integral of 5^xcsc^2(5^x+2) ?

The integral of 5^xcsc^2(5^x+2) is -1/(ln(5))cot(5^x+2)+C

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integral of 5^xcsc^2(5^x+2)

\int 5^{x} csc^{2} (5^{x} + 2) d x

- \frac{1}{ln ( 5 )} cot (5^{x} + 2) + C

Solution steps

\int 5^{x} csc^{2} (5^{x} + 2) d x

Apply u-substitution

: \int \frac{csc ^{2} ( u )}{ln ( 5 )} d u

= \int \frac{csc ^{2} ( u )}{ln ( 5 )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{ln ( 5 )} \cdot \int csc^{2} (u) d u

Use the common integral:

\int csc^{2} (u) d u = - cot (u)

= \frac{1}{ln ( 5 )} (- cot (u))

Substitute back

u = 5^{x} + 2

= \frac{1}{ln ( 5 )} (- cot (5^{x} + 2))

Simplify

= - \frac{1}{ln ( 5 )} cot (5^{x} + 2)

Add a constant to the solution

= - \frac{1}{ln ( 5 )} cot (5^{x} + 2) + C

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What is the integral of 5^xcsc^2(5^x+2) ?
The integral of 5^xcsc^2(5^x+2) is -1/(ln(5))cot(5^x+2)+C