What is the integral of (3-18x^2)csc^2(2x^3-x) ?

The integral of (3-18x^2)csc^2(2x^3-x) is 3cot(2x^3-x)+C

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integral of (3-18x^2)csc^2(2x^3-x)

\int (3 - 18 x^{2}) csc^{2} (2 x^{3} - x) d x

3 cot (2 x^{3} - x) + C

Solution steps

\int (3 - 18 x^{2}) csc^{2} (2 x^{3} - x) d x

Apply u-substitution

: \int - 3 csc^{2} (u) d u

= \int - 3 csc^{2} (u) d u

Take the constant out:

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

= - 3 \cdot \int csc^{2} (u) d u

Use the common integral:

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

= - 3 (- cot (u))

Substitute back

u = 2 x^{3} - x

= - 3 (- cot (2 x^{3} - x))

Simplify

= 3 cot (2 x^{3} - x)

Add a constant to the solution

= 3 cot (2 x^{3} - x) + C

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What is the integral of (3-18x^2)csc^2(2x^3-x) ?
The integral of (3-18x^2)csc^2(2x^3-x) is 3cot(2x^3-x)+C