What is the integral of t/((sin(t^2))^2) ?

The integral of t/((sin(t^2))^2) is -1/2 cot(t^2)+C

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integral of t/((sin(t^2))^2)

\int \frac{t}{( sin ( t ^{2} ) ) ^{2}} d t

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

Solution steps

\int \frac{t}{( sin ( t ^{2} ) ) ^{2}} d t

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

= \int t csc^{2} (t^{2}) d t

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

u = t^{2}

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

Simplify

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

Add a constant to the solution

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

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