What is the integral of (csc^2(sqrt(x)))/(2sqrt(x)) ?

The integral of (csc^2(sqrt(x)))/(2sqrt(x)) is -cot(sqrt(x))+C

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integral of (csc^2(sqrt(x)))/(2sqrt(x))

\int \frac{csc ^{2} ( x )}{2 x} d x

- cot (x) + C

Solution steps

\int \frac{csc ^{2} ( x )}{2 x} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{csc ^{2} ( x )}{x} d x

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

u = x

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

Simplify

\frac{1}{2} \cdot 2 (- cot (x)) : - cot (x)

= - cot (x)

Add a constant to the solution

= - cot (x) + C

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\frac{d}{d x} (9 - x^{\frac{2}{3}})

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f (x) = \frac{2}{x}, at x = - 4

\int \frac{x ^{3}}{x ^{4} - 5} d x

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