What is the integral of xcot^2(2x^2) ?

The integral of xcot^2(2x^2) is 1/4 (-2x^2-cot(2x^2))+C

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integral of xcot^2(2x^2)

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

\frac{1}{4} (- 2 x^{2} - cot (2 x^{2})) + C

Solution steps

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

Apply u-substitution

: \int \frac{cot ^{2} ( u )}{4} d u

= \int \frac{cot ^{2} ( u )}{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 cot^{2} (u) d u

Rewrite using trig identities

= \frac{1}{4} \cdot \int - 1 + csc^{2} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{4} (- \int 1 d u + \int csc^{2} (u) d u)

\int 1 d u = u

\int csc^{2} (u) d u = - cot (u)

= \frac{1}{4} (- u - cot (u))

Substitute back

u = 2 x^{2}

= \frac{1}{4} (- 2 x^{2} - cot (2 x^{2}))

Add a constant to the solution

= \frac{1}{4} (- 2 x^{2} - cot (2 x^{2})) + C

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What is the integral of xcot^2(2x^2) ?
The integral of xcot^2(2x^2) is 1/4 (-2x^2-cot(2x^2))+C