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

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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

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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