What is the integral of 5^xsin(5^x) ?

The integral of 5^xsin(5^x) is -1/(ln(5))cos(5^x)+C

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integral of 5^xsin(5^x)

\int 5^{x} sin (5^{x}) d x

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

Solution steps

\int 5^{x} sin (5^{x}) d x

Apply u-substitution

= \int \frac{sin ( 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 sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

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

Substitute back

u = 5^{x}

= \frac{1}{ln ( 5 )} (- cos (5^{x}))

Simplify

= - \frac{1}{ln ( 5 )} cos (5^{x})

Add a constant to the solution

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

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