What is the integral of 5cot(2x)csc^2(2x) ?

The integral of 5cot(2x)csc^2(2x) is -5/4 cot^2(2x)+C

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integral of 5cot(2x)csc^2(2x)

\int 5 cot (2 x) csc^{2} (2 x) d x

- \frac{5}{4} cot^{2} (2 x) + C

Solution steps

\int 5 cot (2 x) csc^{2} (2 x) d x

Take the constant out:

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

= 5 \cdot \int cot (2 x) csc^{2} (2 x) d x

Apply u-substitution

: \int - \frac{u}{2} d u

= 5 \cdot \int - \frac{u}{2} d u

Take the constant out:

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

= 5 (- \frac{1}{2} \cdot \int u d u)

Apply the Power Rule

: \frac{u ^{2}}{2}

= 5 (- \frac{1}{2} \cdot \frac{u ^{2}}{2})

Substitute back

u = cot (2 x)

= 5 (- \frac{1}{2} \cdot \frac{cot ^{2} ( 2 x )}{2})

Simplify

5 (- \frac{1}{2} \cdot \frac{cot ^{2} ( 2 x )}{2}) : - \frac{5}{4} cot^{2} (2 x)

= - \frac{5}{4} cot^{2} (2 x)

Add a constant to the solution

= - \frac{5}{4} cot^{2} (2 x) + C

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What is the integral of 5cot(2x)csc^2(2x) ?
The integral of 5cot(2x)csc^2(2x) is -5/4 cot^2(2x)+C