What is the integral of 5^{ln(x)} ?

The integral of 5^{ln(x)} is 5^{ln(x)}x-1/(ln(5)+1)ln(5)x^{ln(5)+1}+C

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integral of 5^{ln(x)}

\int 5^{l n (x)} d x

5^{l n (x)} x - \frac{1}{ln ( 5 ) + 1} ln (5) x^{l n (5) + 1} + C

Solution steps

\int 5^{l n (x)} d x

Apply Integration By Parts

= 5^{l n (x)} x - \int ln (5) x^{l n (5)} d x

\int ln (5) x^{l n (5)} d x = \frac{1}{ln ( 5 ) + 1} ln (5) x^{l n (5) + 1}

= 5^{l n (x)} x - \frac{1}{ln ( 5 ) + 1} ln (5) x^{l n (5) + 1}

Add a constant to the solution

= 5^{l n (x)} x - \frac{1}{ln ( 5 ) + 1} ln (5) x^{l n (5) + 1} + C

What is the integral of 5^{ln(x)} ?
The integral of 5^{ln(x)} is 5^{ln(x)}x-1/(ln(5)+1)ln(5)x^{ln(5)+1}+C