What is the integral of (tan(x)+sin(x))/(sec(x)) ?

The integral of (tan(x)+sin(x))/(sec(x)) is -cos(x)+(sin^2(x))/2+C

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integral of (tan(x)+sin(x))/(sec(x))

\int \frac{tan ( x ) + sin ( x )}{sec ( x )} d x

- cos (x) + \frac{sin ^{2} ( x )}{2} + C

Solution steps

\int \frac{tan ( x ) + sin ( x )}{sec ( x )} d x

Expand

\frac{tan ( x ) + sin ( x )}{sec ( x )} : \frac{tan ( x )}{sec ( x )} + \frac{sin ( x )}{sec ( x )}

= \int \frac{tan ( x )}{sec ( x )} + \frac{sin ( 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 \frac{tan ( x )}{sec ( x )} d x + \int \frac{sin ( x )}{sec ( x )} d x

\int \frac{tan ( x )}{sec ( x )} d x = - cos (x)

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

= - cos (x) + \frac{sin ^{2} ( x )}{2}

Add a constant to the solution

= - cos (x) + \frac{sin ^{2} ( x )}{2} + C

\int 6 x e^{5 x} d x

\int x^{2} (5 x^{3} - 1)^{4} d x

\int \frac{x + 1}{x ^{2} - 16} d x

n = 0 \sum \infty 4 n^{2} + 4 n + 1

y = 4 x^{2}, y = x^{2} + 2

What is the integral of (tan(x)+sin(x))/(sec(x)) ?
The integral of (tan(x)+sin(x))/(sec(x)) is -cos(x)+(sin^2(x))/2+C