What is the integral of (2x+1)/((x^2+x+1)^3) ?

The integral of (2x+1)/((x^2+x+1)^3) is -1/(2(x^2+x+1)^2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2x+1)/((x^2+x+1)^3)

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

- \frac{1}{2 ( x ^{2} + x + 1 ) ^{2}} + C

Solution steps

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

Apply u-substitution

: \int \frac{32 u}{( u ^{2} + 3 ) ^{3}} d u

= \int \frac{32 u}{( u ^{2} + 3 ) ^{3}} d u

Take the constant out:

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

= 32 \cdot \int \frac{u}{( u ^{2} + 3 ) ^{3}} d u

Apply u-substitution

: \int \frac{1}{2 v ^{3}} d v

= 32 \cdot \int \frac{1}{2 v ^{3}} d v

Take the constant out:

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

= 32 \cdot \frac{1}{2} \cdot \int \frac{1}{v ^{3}} d v

Apply the Power Rule

: - \frac{1}{2 v ^{2}}

= 32 \cdot \frac{1}{2} (- \frac{1}{2 v ^{2}})

Substitute back

= 32 \cdot \frac{1}{2} - \frac{1}{2 ( ( 2 x + 1 ) ^{2} + 3 ) ^{2}}

Simplify

32 \cdot \frac{1}{2} - \frac{1}{2 ( ( 2 x + 1 ) ^{2} + 3 ) ^{2}} : - \frac{1}{2 ( x ^{2} + x + 1 ) ^{2}}

= - \frac{1}{2 ( x ^{2} + x + 1 ) ^{2}}

Add a constant to the solution

= - \frac{1}{2 ( x ^{2} + x + 1 ) ^{2}} + C

\int \frac{1}{( 5 - x ^{2} ) ^{3}} d x

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

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

\int \frac{16 - t ^{2}}{t ^{2}} d t

\int \frac{7 ln ( x )}{x} d x

What is the integral of (2x+1)/((x^2+x+1)^3) ?
The integral of (2x+1)/((x^2+x+1)^3) is -1/(2(x^2+x+1)^2)+C