What is the integral of (2x+4)/(1+(x+2)^2) ?

The integral of (2x+4)/(1+(x+2)^2) is ln|x^2+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 (2x+4)/(1+(x+2)^2)

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

ln x^{2} + 4 x + 5 + C

Solution steps

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

Apply u-substitution

: \int \frac{2 u}{1 + u ^{2}} d u

= \int \frac{2 u}{1 + u ^{2}} d u

Take the constant out:

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

= 2 \cdot \int \frac{u}{1 + u ^{2}} d u

Apply u-substitution

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

= 2 \cdot \int \frac{1}{2 v} d v

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 \cdot \frac{1}{2} ln ∣ v ∣

Substitute back

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

Simplify

2 \cdot \frac{1}{2} ln 1 + (x + 2)^{2} : ln x^{2} + 4 x + 5

= ln x^{2} + 4 x + 5

Add a constant to the solution

= ln x^{2} + 4 x + 5 + C

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

\int \frac{x + 3}{x - 3} d x

\int (x^{3} - 3 x^{2} + 6 x - 20) d x

\int 2 y^{2} x d x

\int \frac{9}{x ^{2} - 81} d x

What is the integral of (2x+4)/(1+(x+2)^2) ?
The integral of (2x+4)/(1+(x+2)^2) is ln|x^2+4x+5|+C