What is the integral of a^xcos(a^x) ?

The integral of a^xcos(a^x) is 1/(ln(a))sin(a^x)+C

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integral of a^xcos(a^x)

\int a^{x} cos (a^{x}) d x

\frac{1}{ln ( a )} sin (a^{x}) + C

Solution steps

\int a^{x} cos (a^{x}) d x

Apply u-substitution

: \int \frac{cos ( u )}{ln ( a )} d u

= \int \frac{cos ( u )}{ln ( a )} d u

Take the constant out:

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

= \frac{1}{ln ( a )} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{ln ( a )} sin (u)

Substitute back

u = a^{x}

= \frac{1}{ln ( a )} sin (a^{x})

Add a constant to the solution

= \frac{1}{ln ( a )} sin (a^{x}) + C

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What is the integral of a^xcos(a^x) ?
The integral of a^xcos(a^x) is 1/(ln(a))sin(a^x)+C