What is the integral of tan^4(3/5 bx) ?

The integral of tan^4(3/5 bx) is 5/(3b)((tan^3(3/5 bx))/3-(-3/5 bx+tan(3/5 bx)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of tan^4(3/5 bx)

\int tan^{4} (\frac{3}{5} b x) d x

\frac{5}{3 b} (\frac{tan ^{3} ( \frac{3}{5} b x )}{3} - (- \frac{3}{5} b x + tan (\frac{3}{5} b x))) + C

Solution steps

\int tan^{4} (\frac{3}{5} b x) d x

Apply u-substitution

= \int \frac{5 tan ^{4} ( u )}{3 b} d u

Take the constant out:

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

= \frac{5}{3 b} \cdot \int tan^{4} (u) d u

Apply Integral Reduction

= \frac{5}{3 b} (\frac{tan ^{3} ( u )}{3} - \int tan^{2} (u) d u)

\int tan^{2} (u) d u = - u + tan (u)

= \frac{5}{3 b} (\frac{tan ^{3} ( u )}{3} - (- u + tan (u)))

Substitute back

u = \frac{3}{5} b x

= \frac{5}{3 b} (\frac{tan ^{3} ( \frac{3}{5} b x )}{3} - (- \frac{3}{5} b x + tan (\frac{3}{5} b x)))

Add a constant to the solution

= \frac{5}{3 b} (\frac{tan ^{3} ( \frac{3}{5} b x )}{3} - (- \frac{3}{5} b x + tan (\frac{3}{5} b x))) + C

What is the integral of tan^4(3/5 bx) ?
The integral of tan^4(3/5 bx) is 5/(3b)((tan^3(3/5 bx))/3-(-3/5 bx+tan(3/5 bx)))+C