What is the integral of (e^{5sin(x)})/(sec(x)) ?

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

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integral of (e^{5sin(x)})/(sec(x))

\int \frac{e ^{5 s i n (x)}}{sec ( x )} d x

\frac{1}{5} e^{5 s i n (x)} + C

Solution steps

\int \frac{e ^{5 s i n (x)}}{sec ( x )} d x

Use the basic trigonometric identity:

\frac{1}{sec ( x )} = cos (x)

= \int e^{5 s i n (x)} cos (x) d x

Apply u-substitution

: \int \frac{e ^{u}}{5} d u

= \int \frac{e ^{u}}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int e^{u} d u

Use the common integral:

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

= \frac{1}{5} e^{u}

Substitute back

u = 5 sin (x)

= \frac{1}{5} e^{5 s i n (x)}

Add a constant to the solution

= \frac{1}{5} e^{5 s i n (x)} + C

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