What is the integral of 30csc(5x)sec(5x) ?

The integral of 30csc(5x)sec(5x) is 3ln(tan^2(5x))+C

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integral of 30csc(5x)sec(5x)

\int 30 csc (5 x) sec (5 x) d x

3 ln (tan^{2} (5 x)) + C

Solution steps

\int 30 csc (5 x) sec (5 x) d x

Take the constant out:

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

= 30 \cdot \int csc (5 x) sec (5 x) d x

Apply u-substitution

: \int csc (u) sec (u) \frac{1}{5} d u

= 30 \cdot \int csc (u) sec (u) \frac{1}{5} d u

Take the constant out:

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

= 30 \cdot \frac{1}{5} \cdot \int csc (u) sec (u) d u

Apply u-substitution

: \int \frac{v}{v ^{2} - 1} d v

= 30 \cdot \frac{1}{5} \cdot \int \frac{v}{v ^{2} - 1} d v

Apply u-substitution

: \int \frac{1}{2 w} d w

= 30 \cdot \frac{1}{5} \cdot \int \frac{1}{2 w} d w

Take the constant out:

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

= 30 \cdot \frac{1}{5} \cdot \frac{1}{2} \cdot \int \frac{1}{w} d w

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= 30 \cdot \frac{1}{5} \cdot \frac{1}{2} ln ∣ w ∣

Substitute back

= 30 \cdot \frac{1}{5} \cdot \frac{1}{2} ln sec^{2} (5 x) - 1

Simplify

30 \cdot \frac{1}{5} \cdot \frac{1}{2} ln sec^{2} (5 x) - 1 : 3 ln (tan^{2} (5 x))

= 3 ln (tan^{2} (5 x))

Add a constant to the solution

= 3 ln (tan^{2} (5 x)) + C

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What is the integral of 30csc(5x)sec(5x) ?
The integral of 30csc(5x)sec(5x) is 3ln(tan^2(5x))+C