What is the integral of 5csc^2(7w+5) ?

The integral of 5csc^2(7w+5) is -5/7 cot(7w+5)+C

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integral of 5csc^2(7w+5)

\int 5 csc^{2} (7 w + 5) d w

- \frac{5}{7} cot (7 w + 5) + C

Solution steps

\int 5 csc^{2} (7 w + 5) d w

Take the constant out:

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

= 5 \cdot \int csc^{2} (7 w + 5) d w

Apply u-substitution

: \int csc^{2} (u) \frac{1}{7} d u

= 5 \cdot \int csc^{2} (u) \frac{1}{7} d u

Take the constant out:

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

= 5 \cdot \frac{1}{7} \cdot \int csc^{2} (u) d u

Use the common integral:

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

= 5 \cdot \frac{1}{7} (- cot (u))

Substitute back

u = 7 w + 5

= 5 \cdot \frac{1}{7} (- cot (7 w + 5))

Simplify

5 \cdot \frac{1}{7} (- cot (7 w + 5)) : - \frac{5}{7} cot (7 w + 5)

= - \frac{5}{7} cot (7 w + 5)

Add a constant to the solution

= - \frac{5}{7} cot (7 w + 5) + C

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What is the integral of 5csc^2(7w+5) ?
The integral of 5csc^2(7w+5) is -5/7 cot(7w+5)+C