What is the integral of sec^2(x/2)tan^2(x/2) ?

The integral of sec^2(x/2)tan^2(x/2) is 2/3 tan^3(x/2)+C

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integral of sec^2(x/2)tan^2(x/2)

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

\frac{2}{3} tan^{3} (\frac{x}{2}) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1 - cos ^{2} ( \frac{x}{2} )}{cos ^{4} ( \frac{x}{2} )} d x

Expand

\frac{1 - cos ^{2} ( \frac{x}{2} )}{cos ^{4} ( \frac{x}{2} )} : \frac{1}{cos ^{4} ( \frac{x}{2} )} - \frac{1}{cos ^{2} ( \frac{x}{2} )}

= \int \frac{1}{cos ^{4} ( \frac{x}{2} )} - \frac{1}{cos ^{2} ( \frac{x}{2} )} 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{1}{cos ^{4} ( \frac{x}{2} )} d x - \int \frac{1}{cos ^{2} ( \frac{x}{2} )} d x

\int \frac{1}{cos ^{4} ( \frac{x}{2} )} d x = 2 (tan (\frac{x}{2}) + \frac{tan ^{3} ( \frac{x}{2} )}{3})

\int \frac{1}{cos ^{2} ( \frac{x}{2} )} d x = 2 tan (\frac{x}{2})

= 2 (tan (\frac{x}{2}) + \frac{tan ^{3} ( \frac{x}{2} )}{3}) - 2 tan (\frac{x}{2})

Simplify

2 (tan (\frac{x}{2}) + \frac{tan ^{3} ( \frac{x}{2} )}{3}) - 2 tan (\frac{x}{2}) : \frac{2}{3} tan^{3} (\frac{x}{2})

= \frac{2}{3} tan^{3} (\frac{x}{2})

Add a constant to the solution

= \frac{2}{3} tan^{3} (\frac{x}{2}) + C

What is the integral of sec^2(x/2)tan^2(x/2) ?
The integral of sec^2(x/2)tan^2(x/2) is 2/3 tan^3(x/2)+C