What is the integral of sec^2(x)e^{3tan(x)} ?

The integral of sec^2(x)e^{3tan(x)} is 1/3 e^{3tan(x)}+C

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integral of sec^2(x)e^{3tan(x)}

\int sec^{2} (x) e^{3 t a n (x)} d x

\frac{1}{3} e^{3 t a n (x)} + C

Solution steps

\int sec^{2} (x) e^{3 t a n (x)} d x

Apply u-substitution

: \int e^{3 u} d u

= \int e^{3 u} d u

Apply u-substitution

: \int e^{v} \frac{1}{3} d v

= \int e^{v} \frac{1}{3} d v

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

= \frac{1}{3} e^{3 t a n (x)}

Add a constant to the solution

= \frac{1}{3} e^{3 t a n (x)} + C

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What is the integral of sec^2(x)e^{3tan(x)} ?
The integral of sec^2(x)e^{3tan(x)} is 1/3 e^{3tan(x)}+C