What is the integral of cot(10x)csc^4(10z) ?

The integral of cot(10x)csc^4(10z) is csc^4(10z) 1/10 ln|sin(10x)|+C

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integral of cot(10x)csc^4(10z)

\int cot (10 x) csc^{4} (10 z)

csc^{4} (10 z) \frac{1}{10} ln ∣ sin (10 x) ∣ + C

Solution steps

\int cot (10 x) csc^{4} (10 z) d x

Take the constant out:

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

= csc^{4} (10 z) \cdot \int cot (10 x) d x

Rewrite using trig identities

= csc^{4} (10 z) \cdot \int \frac{cos ( 10 x )}{sin ( 10 x )} d x

Apply u-substitution

= csc^{4} (10 z) \cdot \int \frac{1}{10 u} d u

Take the constant out:

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

= csc^{4} (10 z) \frac{1}{10} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= csc^{4} (10 z) \frac{1}{10} ln ∣ u ∣

Substitute back

u = sin (10 x)

= csc^{4} (10 z) \frac{1}{10} ln ∣ sin (10 x) ∣

Add a constant to the solution

= csc^{4} (10 z) \frac{1}{10} ln ∣ sin (10 x) ∣ + C

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What is the integral of cot(10x)csc^4(10z) ?
The integral of cot(10x)csc^4(10z) is csc^4(10z) 1/10 ln|sin(10x)|+C