What is the integral of (18)/((4x^2+9)^2) ?

The integral of (18)/((4x^2+9)^2) is 1/6 (arctan(2/3 x)+1/2 sin(2arctan(2/3 x)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (18)/((4x^2+9)^2)

\int \frac{18}{( 4 x ^{2} + 9 ) ^{2}} d x

\frac{1}{6} (arctan (\frac{2}{3} x) + \frac{1}{2} sin (2 arctan (\frac{2}{3} x))) + C

Solution steps

\int \frac{18}{( 4 x ^{2} + 9 ) ^{2}} d x

Take the constant out:

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

= 18 \cdot \int \frac{1}{( 4 x ^{2} + 9 ) ^{2}} d x

Apply Integral Substitution

= 18 \cdot \int \frac{1}{54 ( u ^{2} + 1 ) ^{2}} d u

Take the constant out:

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

= 18 \cdot \frac{1}{54} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Apply Trigonometric Substitution

: \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

= 18 \cdot \frac{1}{54} \cdot \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

Rewrite using trig identities

= 18 \cdot \frac{1}{54} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

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

= 18 \cdot \frac{1}{54} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Apply the Sum Rule:

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

= 18 \cdot \frac{1}{54} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= 18 \cdot \frac{1}{54} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 18 \cdot \frac{1}{54} \cdot \frac{1}{2} (arctan (\frac{2}{3} x) + \frac{1}{2} sin (2 arctan (\frac{2}{3} x)))

Simplify

18 \cdot \frac{1}{54} \cdot \frac{1}{2} (arctan (\frac{2}{3} x) + \frac{1}{2} sin (2 arctan (\frac{2}{3} x))) : \frac{1}{6} (arctan (\frac{2}{3} x) + \frac{1}{2} sin (2 arctan (\frac{2}{3} x)))

= \frac{1}{6} (arctan (\frac{2}{3} x) + \frac{1}{2} sin (2 arctan (\frac{2}{3} x)))

Add a constant to the solution

= \frac{1}{6} (arctan (\frac{2}{3} x) + \frac{1}{2} sin (2 arctan (\frac{2}{3} x))) + C

\int \frac{3 ^{x + 1} - 6 ^{x - 1}}{1 2 ^{x}} d x

\int \frac{3 x + 13}{( x + 1 ) ( x ^{2} + 4 )} d x

\int (3 x^{2} - 2 x - 1) d x

\int \frac{( 1 - x ) e ^{x}}{x ^{2}} d x

\int \frac{1}{( 4 x ^{2} - 1 ) ( x + 7 )} d x

What is the integral of (18)/((4x^2+9)^2) ?
The integral of (18)/((4x^2+9)^2) is 1/6 (arctan(2/3 x)+1/2 sin(2arctan(2/3 x)))+C