What is the integral of 8(tan(8x)) ?

The integral of 8(tan(8x)) is -ln|cos(8x)|+C

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integral of 8(tan(8x))

\int 8 (tan (8 x)) d x

- ln ∣ cos (8 x) ∣ + C

Solution steps

\int 8 tan (8 x) d x

Take the constant out:

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

= 8 \cdot \int tan (8 x) d x

Rewrite using trig identities

= 8 \cdot \int \frac{sin ( 8 x )}{cos ( 8 x )} d x

Apply u-substitution

: \int - \frac{1}{8 u} d u

= 8 \cdot \int - \frac{1}{8 u} d u

Take the constant out:

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

= 8 (- \frac{1}{8} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 8 (- \frac{1}{8} ln ∣ u ∣)

Substitute back

u = cos (8 x)

= 8 (- \frac{1}{8} ln ∣ cos (8 x) ∣)

Simplify

8 (- \frac{1}{8} ln ∣ cos (8 x) ∣) : - ln ∣ cos (8 x) ∣

= - ln ∣ cos (8 x) ∣

Add a constant to the solution

= - ln ∣ cos (8 x) ∣ + C

\int_{- 1}^{1} (1 - x^{2}) d x

f (x) = x^{2} + 1, (4, 17)

h (x) = (x^{2} - 3 x + 2) (2 x^{3} + 1)

x^{3}, at x = - 1

\frac{\partial}{\partial x} (6 y x^{5} - 20 y^{2} x^{4})

What is the integral of 8(tan(8x)) ?
The integral of 8(tan(8x)) is -ln|cos(8x)|+C