What is the integral of 9^xsin(9^x) ?

The integral of 9^xsin(9^x) is -1/(2ln(3))cos(9^x)+C

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integral of 9^xsin(9^x)

\int 9^{x} sin (9^{x}) d x

- \frac{1}{2 ln ( 3 )} cos (9^{x}) + C

Solution steps

\int 9^{x} sin (9^{x}) d x

Apply u-substitution

: \int \frac{sin ( u )}{2 ln ( 3 )} d u

= \int \frac{sin ( u )}{2 ln ( 3 )} d u

Take the constant out:

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

= \frac{1}{2 ln ( 3 )} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{2 ln ( 3 )} (- cos (u))

Substitute back

u = 9^{x}

= \frac{1}{2 ln ( 3 )} (- cos (9^{x}))

Simplify

= - \frac{1}{2 ln ( 3 )} cos (9^{x})

Add a constant to the solution

= - \frac{1}{2 ln ( 3 )} cos (9^{x}) + C

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What is the integral of 9^xsin(9^x) ?
The integral of 9^xsin(9^x) is -1/(2ln(3))cos(9^x)+C