What is the integral of e^xsin(6e^x+3) ?

The integral of e^xsin(6e^x+3) is -1/6 cos(6e^x+3)+C

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integral of e^xsin(6e^x+3)

\int e^{x} sin (6 e^{x} + 3) d x

- \frac{1}{6} cos (6 e^{x} + 3) + C

Solution steps

\int e^{x} sin (6 e^{x} + 3) d x

Apply u-substitution

= \int sin (6 u + 3) d u

Apply u-substitution

= \int sin (v) \frac{1}{6} d v

Take the constant out:

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

= \frac{1}{6} \cdot \int sin (v) d v

Use the common integral:

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

= \frac{1}{6} (- cos (v))

Substitute back

= \frac{1}{6} (- cos (6 e^{x} + 3))

Simplify

= - \frac{1}{6} cos (6 e^{x} + 3)

Add a constant to the solution

= - \frac{1}{6} cos (6 e^{x} + 3) + C

What is the integral of e^xsin(6e^x+3) ?
The integral of e^xsin(6e^x+3) is -1/6 cos(6e^x+3)+C