What is the integral of 6x*csc^2(8x^2) ?

The integral of 6x*csc^2(8x^2) is -3/8 cot(8x^2)+C

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integral of 6x*csc^2(8x^2)

\int 6 x \cdot csc^{2} (8 x^{2}) d x

- \frac{3}{8} cot (8 x^{2}) + C

Solution steps

\int 6 x csc^{2} (8 x^{2}) d x

Take the constant out:

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

= 6 \cdot \int x csc^{2} (8 x^{2}) d x

Apply u-substitution

: \int \frac{csc ^{2} ( 8 u )}{2} d u

= 6 \cdot \int \frac{csc ^{2} ( 8 u )}{2} d u

Take the constant out:

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

= 6 \cdot \frac{1}{2} \cdot \int csc^{2} (8 u) d u

Apply u-substitution

: \int csc^{2} (v) \frac{1}{8} d v

= 6 \cdot \frac{1}{2} \cdot \int csc^{2} (v) \frac{1}{8} d v

Take the constant out:

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

= 6 \cdot \frac{1}{2} \cdot \frac{1}{8} \cdot \int csc^{2} (v) d v

Use the common integral:

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

= 6 \cdot \frac{1}{2} \cdot \frac{1}{8} (- cot (v))

Substitute back

= 6 \cdot \frac{1}{2} \cdot \frac{1}{8} (- cot (8 x^{2}))

Simplify

6 \cdot \frac{1}{2} \cdot \frac{1}{8} (- cot (8 x^{2})) : - \frac{3}{8} cot (8 x^{2})

= - \frac{3}{8} cot (8 x^{2})

Add a constant to the solution

= - \frac{3}{8} cot (8 x^{2}) + C

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What is the integral of 6x*csc^2(8x^2) ?
The integral of 6x*csc^2(8x^2) is -3/8 cot(8x^2)+C