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Popular >

integral of csc^7(4x)

Solution

\int csc^{7} (4 x) d x

Solution

\frac{1}{4} (- \frac{1}{6} csc^{6} (4 x) cos (4 x) + \frac{5}{6} (- \frac{1}{4} csc^{4} (4 x) cos (4 x) + \frac{3}{4} (- \frac{1}{2} cot (4 x) csc (4 x) + \frac{1}{2} ln ∣ tan (2 x) ∣))) + C

Solution steps

\int csc^{7} (4 x) d x

Apply u-substitution

: \int csc^{7} (u) \frac{1}{4} d u

= \int csc^{7} (u) \frac{1}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int csc^{7} (u) d u

Apply Integral Reduction

= \frac{1}{4} (- \frac{cos ( u ) csc ^{6} ( u )}{6} + \frac{5}{6} \cdot \int csc^{5} (u) d u)

\int csc^{5} (u) d u = - \frac{cos ( u ) csc ^{4} ( u )}{4} + \frac{3}{4} (- \frac{1}{2} cot (u) csc (u) + \frac{1}{2} ln tan (\frac{u}{2}))

= \frac{1}{4} (- \frac{cos ( u ) csc ^{6} ( u )}{6} + \frac{5}{6} (- \frac{cos ( u ) csc ^{4} ( u )}{4} + \frac{3}{4} (- \frac{1}{2} cot (u) csc (u) + \frac{1}{2} ln tan (\frac{u}{2}))))

Substitute back

u = 4 x

= \frac{1}{4} (- \frac{cos ( 4 x ) csc ^{6} ( 4 x )}{6} + \frac{5}{6} (- \frac{cos ( 4 x ) csc ^{4} ( 4 x )}{4} + \frac{3}{4} (- \frac{1}{2} cot (4 x) csc (4 x) + \frac{1}{2} ln tan (\frac{4 x}{2}))))

Simplify

\frac{1}{4} (- \frac{cos ( 4 x ) csc ^{6} ( 4 x )}{6} + \frac{5}{6} (- \frac{cos ( 4 x ) csc ^{4} ( 4 x )}{4} + \frac{3}{4} (- \frac{1}{2} cot (4 x) csc (4 x) + \frac{1}{2} ln tan (\frac{4 x}{2})))) : \frac{1}{4} (- \frac{1}{6} csc^{6} (4 x) cos (4 x) + \frac{5}{6} (- \frac{1}{4} csc^{4} (4 x) cos (4 x) + \frac{3}{4} (- \frac{1}{2} cot (4 x) csc (4 x) + \frac{1}{2} ln ∣ tan (2 x) ∣)))

= \frac{1}{4} (- \frac{1}{6} csc^{6} (4 x) cos (4 x) + \frac{5}{6} (- \frac{1}{4} csc^{4} (4 x) cos (4 x) + \frac{3}{4} (- \frac{1}{2} cot (4 x) csc (4 x) + \frac{1}{2} ln ∣ tan (2 x) ∣)))

Add a constant to the solution

= \frac{1}{4} (- \frac{1}{6} csc^{6} (4 x) cos (4 x) + \frac{5}{6} (- \frac{1}{4} csc^{4} (4 x) cos (4 x) + \frac{3}{4} (- \frac{1}{2} cot (4 x) csc (4 x) + \frac{1}{2} ln ∣ tan (2 x) ∣))) + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of csc^7(4x) ?
The integral of csc^7(4x) is