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

The integral of csc^4(x)cos(x) is -1/3 csc^3(x)+C

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integral of csc^4(x)cos(x)

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

- \frac{1}{3} csc^{3} (x) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{cos ( x )}{sin ^{4} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{4}} d u

= \int \frac{1}{u ^{4}} d u

Apply the Power Rule

: - \frac{1}{3 u ^{3}}

= - \frac{1}{3 u ^{3}}

Substitute back

u = sin (x)

= - \frac{1}{3 sin ^{3} ( x )}

Simplify

- \frac{1}{3 sin ^{3} ( x )} : - \frac{1}{3} csc^{3} (x)

= - \frac{1}{3} csc^{3} (x)

Add a constant to the solution

= - \frac{1}{3} csc^{3} (x) + C

\int x^{4} 3 x^{5} - 5 d x

\frac{d}{d x} (\frac{5}{x x})

(z, 2), (- 2, - 6)

\int 6 x sec (3 x^{2}) tan (3 x^{2}) d x

x^{2} y^{^{''}} + x y^{^{'}} + 25 y = 0

What is the integral of csc^4(x)cos(x) ?
The integral of csc^4(x)cos(x) is -1/3 csc^3(x)+C