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Popular Calculus >

integral of sec^5(x/2)

Solution

\int sec^{5} (\frac{x}{2}) d x

Solution

2 (\frac{sec ^{4} ( \frac{x}{2} ) sin ( \frac{x}{2} )}{4} + \frac{3}{4} (\frac{1}{2} sec (\frac{x}{2}) tan (\frac{x}{2}) + \frac{1}{2} ln tan (\frac{x}{2}) + sec (\frac{x}{2}))) + C

Solution steps

\int sec^{5} (\frac{x}{2}) d x

Apply u-substitution

= \int sec^{5} (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot \int sec^{5} (u) d u

Apply Integral Reduction

= 2 (\frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} \cdot \int sec^{3} (u) d u)

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= 2 (\frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣))

Substitute back

u = \frac{x}{2}

= 2 (\frac{sec ^{4} ( \frac{x}{2} ) sin ( \frac{x}{2} )}{4} + \frac{3}{4} (\frac{1}{2} sec (\frac{x}{2}) tan (\frac{x}{2}) + \frac{1}{2} ln tan (\frac{x}{2}) + sec (\frac{x}{2})))

Add a constant to the solution

= 2 (\frac{sec ^{4} ( \frac{x}{2} ) sin ( \frac{x}{2} )}{4} + \frac{3}{4} (\frac{1}{2} sec (\frac{x}{2}) tan (\frac{x}{2}) + \frac{1}{2} ln tan (\frac{x}{2}) + sec (\frac{x}{2}))) + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of sec^5(x/2) ?
The integral of sec^5(x/2) is