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

The integral of sec^2(x)tan(x)e^{sec(x)} is -e^{sec(x)}+e^{sec(x)}ln(e^{sec(x)})+C

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

\int sec^{2} (x) tan (x) e^{s e c (x)} d x

- e^{s e c (x)} + e^{s e c (x)} ln (e^{s e c (x)}) + C

Solution steps

\int sec^{2} (x) tan (x) e^{s e c (x)} d x

Apply u-substitution

: \int e^{1 + u^{2}} u d u

= \int e^{1 + u^{2}} u d u

Apply u-substitution

: \int ln (v) d v

= \int ln (v) d v

Apply Integration By Parts

: v ln (v) - \int 1 d v

= v ln (v) - \int 1 d v

\int 1 d v = v

= v ln (v) - v

Substitute back

= e^{1 + t a n^{2} (x)} ln (e^{1 + t a n^{2} (x)}) - e^{1 + t a n^{2} (x)}

Simplify

e^{1 + t a n^{2} (x)} ln (e^{1 + t a n^{2} (x)}) - e^{1 + t a n^{2} (x)} : - e^{s e c (x)} + e^{s e c (x)} ln (e^{s e c (x)})

= - e^{s e c (x)} + e^{s e c (x)} ln (e^{s e c (x)})

Add a constant to the solution

= - e^{s e c (x)} + e^{s e c (x)} ln (e^{s e c (x)}) + C

sin (cos (5 x))

\int sin (x) + 8 d x

\int \frac{8}{x ^{8}} d x

b x^{4}

\frac{d}{d x} ((1 - x)^{4})

What is the integral of sec^2(x)tan(x)e^{sec(x)} ?
The integral of sec^2(x)tan(x)e^{sec(x)} is -e^{sec(x)}+e^{sec(x)}ln(e^{sec(x)})+C