What is the integral of (sin(x))(sec^2(x)+cot^2(x)) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sin(x))(sec^2(x)+cot^2(x))

\int (sin (x)) (sec^{2} (x) + cot^{2} (x)) d x

sec (x) + ln tan (\frac{x}{2}) + cos (x) + C

Solution steps

\int sin (x) (sec^{2} (x) + cot^{2} (x)) d x

Rewrite using trig identities

= \int \frac{sin ^{2} ( x ) + cos ^{4} ( x )}{cos ^{2} ( x ) sin ( x )} d x

Expand

\frac{sin ^{2} ( x ) + cos ^{4} ( x )}{cos ^{2} ( x ) sin ( x )} : \frac{sin ( x )}{cos ^{2} ( x )} + \frac{cos ^{2} ( x )}{sin ( x )}

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

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

\int \frac{cos ^{2} ( x )}{sin ( x )} d x = ln tan (\frac{x}{2}) + cos (x)

= sec (x) + ln tan (\frac{x}{2}) + cos (x)

Add a constant to the solution

= sec (x) + ln tan (\frac{x}{2}) + cos (x) + C

\int \frac{3}{3 x - 5} d x

\int 4^{5 x} d x

\int \frac{x}{5 - 4 x} d x

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

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

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