What is the integral of (cos(x)+sin^3(x))/(sec(x)) ?

The integral of (cos(x)+sin^3(x))/(sec(x)) is 1/2 (x+1/2 sin(2x))+(sin^4(x))/4+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (cos(x)+sin^3(x))/(sec(x))

\int \frac{cos ( x ) + sin ^{3} ( x )}{sec ( x )} d x

\frac{1}{2} (x + \frac{1}{2} sin (2 x)) + \frac{sin ^{4} ( x )}{4} + C

Solution steps

\int \frac{cos ( x ) + sin ^{3} ( x )}{sec ( x )} d x

Expand

\frac{cos ( x ) + sin ^{3} ( x )}{sec ( x )} : \frac{cos ( x )}{sec ( x )} + \frac{sin ^{3} ( x )}{sec ( x )}

= \int \frac{cos ( x )}{sec ( x )} + \frac{sin ^{3} ( 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{cos ( x )}{sec ( x )} d x + \int \frac{sin ^{3} ( x )}{sec ( x )} d x

\int \frac{cos ( x )}{sec ( x )} d x = \frac{1}{2} (x + \frac{1}{2} sin (2 x))

\int \frac{sin ^{3} ( x )}{sec ( x )} d x = \frac{sin ^{4} ( x )}{4}

= \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + \frac{sin ^{4} ( x )}{4}

Add a constant to the solution

= \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + \frac{sin ^{4} ( x )}{4} + C

\int (5 x^{2} - 2 sin (x)) d x

\int \frac{( x ^{2} - 5 x + 6 )}{x ^{2} + 4} d x

\int 3 sec^{2} (2 t) d t

\int \frac{2}{e ^{x} - e ^{- x}} d x

\int 2 ln (x) x^{4} d x

What is the integral of (cos(x)+sin^3(x))/(sec(x)) ?
The integral of (cos(x)+sin^3(x))/(sec(x)) is 1/2 (x+1/2 sin(2x))+(sin^4(x))/4+C