What is the integral of e^{7x}cos(e^{7x}+5) ?

The integral of e^{7x}cos(e^{7x}+5) is 1/7 sin(e^{7x}+5)+C

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integral of e^{7x}cos(e^{7x}+5)

\int e^{7 x} cos (e^{7 x} + 5) d x

\frac{1}{7} sin (e^{7 x} + 5) + C

Solution steps

\int e^{7 x} cos (e^{7 x} + 5) d x

Apply u-substitution

: \int e^{u} cos (e^{u} + 5) \frac{1}{7} d u

= \int e^{u} cos (e^{u} + 5) \frac{1}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int e^{u} cos (e^{u} + 5) d u

Apply u-substitution

: \int cos (v + 5) d v

= \frac{1}{7} \cdot \int cos (v + 5) d v

Apply u-substitution

: \int cos (w) d w

= \frac{1}{7} \cdot \int cos (w) d w

Use the common integral:

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

= \frac{1}{7} sin (w)

Substitute back

= \frac{1}{7} sin (e^{7 x} + 5)

Add a constant to the solution

= \frac{1}{7} sin (e^{7 x} + 5) + C

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What is the integral of e^{7x}cos(e^{7x}+5) ?
The integral of e^{7x}cos(e^{7x}+5) is 1/7 sin(e^{7x}+5)+C