What is the integral of (16csc(8x)sec(8x)) ?

The integral of (16csc(8x)sec(8x)) is ln(tan^2(8x))+C

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integral of (16csc(8x)sec(8x))

\int (16 csc (8 x) sec (8 x)) d x

ln (tan^{2} (8 x)) + C

Solution steps

\int 16 csc (8 x) sec (8 x) d x

Take the constant out:

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

= 16 \cdot \int csc (8 x) sec (8 x) d x

Apply u-substitution

= 16 \cdot \int csc (u) sec (u) \frac{1}{8} d u

Take the constant out:

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

= 16 \cdot \frac{1}{8} \cdot \int csc (u) sec (u) d u

Apply u-substitution

= 16 \cdot \frac{1}{8} \cdot \int \frac{v}{v ^{2} - 1} d v

Apply u-substitution

= 16 \cdot \frac{1}{8} \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

= 16 \cdot \frac{1}{8} \cdot \frac{1}{2} \cdot \int \frac{1}{w} d w

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= 16 \cdot \frac{1}{8} \cdot \frac{1}{2} ln ∣ w ∣

Substitute back

= 16 \cdot \frac{1}{8} \cdot \frac{1}{2} ln sec^{2} (8 x) - 1

Simplify

16 \cdot \frac{1}{8} \cdot \frac{1}{2} ln sec^{2} (8 x) - 1 : ln (tan^{2} (8 x))

= ln (tan^{2} (8 x))

Add a constant to the solution

= ln (tan^{2} (8 x)) + C

What is the integral of (16csc(8x)sec(8x)) ?
The integral of (16csc(8x)sec(8x)) is ln(tan^2(8x))+C