What is the integral of (e^{3sin(x)})/(sec(x)) ?

The integral of (e^{3sin(x)})/(sec(x)) is 1/3 e^{3sin(x)}+C

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integral of (e^{3sin(x)})/(sec(x))

\int \frac{e ^{3 s i n (x)}}{sec ( x )} d x

\frac{1}{3} e^{3 s i n (x)} + C

Solution steps

\int \frac{e ^{3 s i n (x)}}{sec ( x )} d x

Use the basic trigonometric identity:

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

= \int e^{3 s i n (x)} cos (x) d x

Apply u-substitution

: \int \frac{e ^{u}}{3} d u

= \int \frac{e ^{u}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int e^{u} d u

Use the common integral:

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

= \frac{1}{3} e^{u}

Substitute back

u = 3 sin (x)

= \frac{1}{3} e^{3 s i n (x)}

Add a constant to the solution

= \frac{1}{3} e^{3 s i n (x)} + C

f (x) = x^{2 x}

\int \frac{- ln ( x )}{x} d x

y = c e^{x}

\int \frac{1}{cot ( y )} d y

x \to 0 - lim (x^{2} - 1)

What is the integral of (e^{3sin(x)})/(sec(x)) ?
The integral of (e^{3sin(x)})/(sec(x)) is 1/3 e^{3sin(x)}+C