What is the integral of e^{6cos(6x)}(-6sin(6x)) ?

The integral of e^{6cos(6x)}(-6sin(6x)) is 1/6 e^{6cos(6x)}+C

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integral of e^{6cos(6x)}(-6sin(6x))

\int e^{6 c o s (6 x)} (- 6 sin (6 x)) d x

\frac{1}{6} e^{6 c o s (6 x)} + C

Solution steps

\int e^{6 c o s (6 x)} (- 6 sin (6 x)) d x

Simplify

= \int - 6 e^{6 c o s (6 x)} sin (6 x) d x

Take the constant out:

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

= - 6 \cdot \int e^{6 c o s (6 x)} sin (6 x) d x

Apply u-substitution

: \int - \frac{e ^{u}}{36} d u

= - 6 \cdot \int - \frac{e ^{u}}{36} d u

Take the constant out:

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

= - 6 (- \frac{1}{36} \cdot \int e^{u} d u)

Use the common integral:

\int e^{u} d u = e^{u}

= - 6 (- \frac{1}{36} e^{u})

Substitute back

u = 6 cos (6 x)

= - 6 (- \frac{1}{36} e^{6 c o s (6 x)})

Simplify

- 6 (- \frac{1}{36} e^{6 c o s (6 x)}) : \frac{1}{6} e^{6 c o s (6 x)}

= \frac{1}{6} e^{6 c o s (6 x)}

Add a constant to the solution

= \frac{1}{6} e^{6 c o s (6 x)} + C

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What is the integral of e^{6cos(6x)}(-6sin(6x)) ?
The integral of e^{6cos(6x)}(-6sin(6x)) is 1/6 e^{6cos(6x)}+C