What is the integral of 2sec^2(4/5 x) ?

The integral of 2sec^2(4/5 x) is 5/2 tan((4x)/5)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 2sec^2(4/5 x)

\int 2 sec^{2} (\frac{4}{5} x) d x

\frac{5}{2} tan (\frac{4 x}{5}) + C

Solution steps

\int 2 sec^{2} (\frac{4}{5} x) d x

Take the constant out:

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

= 2 \cdot \int sec^{2} (\frac{4}{5} x) d x

Rewrite using trig identities

= 2 \cdot \int \frac{1}{cos ^{2} ( \frac{4 x}{5} )} d x

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= 2 \cdot \int sec^{2} (\frac{4 x}{5}) d x

Rewrite using trig identities

= 2 \cdot \int 1 + tan^{2} (\frac{4 x}{5}) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int 1 d x + \int tan^{2} (\frac{4 x}{5}) d x)

\int 1 d x = x

\int tan^{2} (\frac{4 x}{5}) d x = - x + \frac{5}{4} tan (\frac{4 x}{5})

= 2 (x - x + \frac{5}{4} tan (\frac{4 x}{5}))

Simplify

2 (x - x + \frac{5}{4} tan (\frac{4 x}{5})) : \frac{5}{2} tan (\frac{4 x}{5})

= \frac{5}{2} tan (\frac{4 x}{5})

Add a constant to the solution

= \frac{5}{2} tan (\frac{4 x}{5}) + C

\int cos (x) \frac{1}{x} d x

\int x cos (x) x d x

\int (2 - 7 x)^{\frac{2}{3}} d x

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

\int \frac{cos ^{3} ( 3 x )}{sin ^{4} ( 3 x )} d x

What is the integral of 2sec^2(4/5 x) ?
The integral of 2sec^2(4/5 x) is 5/2 tan((4x)/5)+C