What is the integral of 1/(sin^2(2x+pi/4)) ?

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

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integral of 1/(sin^2(2x+pi/4))

\int \frac{1}{sin ^{2} ( 2 x + \frac{π}{4} )} d x

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

Solution steps

\int \frac{1}{sin ^{2} ( 2 x + \frac{π}{4} )} d x

Apply u-substitution

: \int \frac{1}{2 sin ^{2} ( u )} d u

= \int \frac{1}{2 sin ^{2} ( u )} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{sin ^{2} ( u )} d u

Use the common integral:

\int \frac{1}{sin ^{2} ( u )} d u = - cot (u)

= \frac{1}{2} (- cot (u))

Substitute back

u = 2 x + \frac{π}{4}

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

Simplify

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

Add a constant to the solution

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

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