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

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

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

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

tan (x) + e^{s i n (x)} + C

Solution steps

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

Expand

\frac{sec ^{3} ( x ) + e ^{s i n (x)}}{sec ( x )} : sec^{2} (x) + \frac{e ^{s i n (x)}}{sec ( x )}

= \int sec^{2} (x) + \frac{e ^{s i n (x)}}{sec ( x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int sec^{2} (x) d x + \int \frac{e ^{s i n (x)}}{sec ( x )} d x

\int sec^{2} (x) d x = tan (x)

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

= tan (x) + e^{s i n (x)}

Add a constant to the solution

= tan (x) + e^{s i n (x)} + C

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