What is the integral of tan(4θ) ?

The integral of tan(4θ) is -1/4 ln|cos(4θ)|+C

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integral of tan(4θ)

\int tan (4 θ) d θ

- \frac{1}{4} ln ∣ cos (4 θ) ∣ + C

Solution steps

\int tan (4 θ) d θ

Rewrite using trig identities

= \int \frac{sin ( 4 θ )}{cos ( 4 θ )} d θ

Apply u-substitution

: \int - \frac{1}{4 u} d u

= \int - \frac{1}{4 u} d u

Take the constant out:

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

= - \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{4} ln ∣ u ∣

Substitute back

u = cos (4 θ)

= - \frac{1}{4} ln ∣ cos (4 θ) ∣

Add a constant to the solution

= - \frac{1}{4} ln ∣ cos (4 θ) ∣ + C

f (x) = 4 e^{x}

x \to - \infty lim (∣ x ∣ - 3)

(x + 1) ln (x + 1)

y = (\frac{4}{x + 2}), at x = 2

\int \frac{6}{7} sec^{2} (\frac{x}{7}) d x