What is the integral of tan^3((pix)/2)sec^2((pix)/2) ?

The integral of tan^3((pix)/2)sec^2((pix)/2) is 1/(2pi)tan^4((pix)/2)+C

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integral of tan^3((pix)/2)sec^2((pix)/2)

\int tan^{3} (\frac{π x}{2}) sec^{2} (\frac{π x}{2}) d x

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

Solution steps

\int tan^{3} (\frac{π x}{2}) sec^{2} (\frac{π x}{2}) d x

Apply u-substitution

= \int tan^{3} (u) sec^{2} (u) \frac{2}{π} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{π} \cdot \int tan^{3} (u) sec^{2} (u) d u

Rewrite using trig identities

= \frac{2}{π} \cdot \int (1 + tan^{2} (u)) tan^{3} (u) d u

Apply u-substitution

= \frac{2}{π} \cdot \int v^{3} d v

Apply the Power Rule

= \frac{2}{π} \cdot \frac{v ^{4}}{4}

Substitute back

= \frac{2}{π} \cdot \frac{tan ^{4} ( \frac{π x}{2} )}{4}

Simplify

\frac{2}{π} \cdot \frac{tan ^{4} ( \frac{π x}{2} )}{4} : \frac{1}{2 π} tan^{4} (\frac{π x}{2})

= \frac{1}{2 π} tan^{4} (\frac{π x}{2})

Add a constant to the solution

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

What is the integral of tan^3((pix)/2)sec^2((pix)/2) ?
The integral of tan^3((pix)/2)sec^2((pix)/2) is 1/(2pi)tan^4((pix)/2)+C