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

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

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integral of e^{cos(7x)}sin(7x)

\int e^{c o s (7 x)} sin (7 x) d x

- \frac{1}{7} e^{c o s (7 x)} + C

Solution steps

\int e^{c o s (7 x)} sin (7 x) d x

Apply u-substitution

: \int e^{c o s (u)} sin (u) \frac{1}{7} d u

= \int e^{c o s (u)} sin (u) \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^{c o s (u)} sin (u) d u

Apply u-substitution

: \int - e^{v} d v

= \frac{1}{7} \cdot \int - e^{v} d v

Take the constant out:

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

= \frac{1}{7} (- \int e^{v} d v)

Use the common integral:

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

= \frac{1}{7} (- e^{v})

Substitute back

= \frac{1}{7} (- e^{c o s (7 x)})

Simplify

= - \frac{1}{7} e^{c o s (7 x)}

Add a constant to the solution

= - \frac{1}{7} e^{c o s (7 x)} + C

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