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Popular Calculus >

integral of sec^5(θ/3)

Solution

\int sec^{5} (\frac{θ}{3}) d θ

Solution

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

Solution steps

\int sec^{5} (\frac{θ}{3}) d θ

Apply u-substitution

= \int sec^{5} (u) \cdot 3 d u

Take the constant out:

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

= 3 \cdot \int sec^{5} (u) d u

Apply Integral Reduction

= 3 (\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) ∣

= 3 (\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{θ}{3}

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

Add a constant to the solution

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

Graph

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Frequently Asked Questions (FAQ)

What is the integral of sec^5(θ/3) ?
The integral of sec^5(θ/3) is