What is the integral of (csc^2(1/x))/(5x^2) ?

The integral of (csc^2(1/x))/(5x^2) is 1/5 cot(1/x)+C

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integral of (csc^2(1/x))/(5x^2)

\int \frac{csc ^{2} ( \frac{1}{x} )}{5 x ^{2}} d x

\frac{1}{5} cot (\frac{1}{x}) + C

Solution steps

\int \frac{csc ^{2} ( \frac{1}{x} )}{5 x ^{2}} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{csc ^{2} ( \frac{1}{x} )}{x ^{2}} d x

Apply u-substitution

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

= \frac{1}{5} \cdot \int - 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}{5} (- \int csc^{2} (u) d u)

Use the common integral:

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

= \frac{1}{5} (- (- cot (u)))

Substitute back

u = \frac{1}{x}

= \frac{1}{5} (- (- cot (\frac{1}{x})))

Simplify

= \frac{1}{5} cot (\frac{1}{x})

Add a constant to the solution

= \frac{1}{5} cot (\frac{1}{x}) + C

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What is the integral of (csc^2(1/x))/(5x^2) ?
The integral of (csc^2(1/x))/(5x^2) is 1/5 cot(1/x)+C