en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of ((x+3))/(sqrt(4x^2+4x+3))

Solution

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

Solution

\frac{1}{4} (5 (ln 2 x + 2 \frac{4 x ^{2} + 4 x + 3}{2} + 1 - \frac{ln ( 2 )}{2}) + 4 x^{2} + 4 x + 3) + C

Solution steps

\int \frac{x + 3}{4 x ^{2} + 4 x + 3} d x

x + 3 = 3 (\frac{x}{3} + 1)

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

Take the constant out:

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

= 3 \cdot \int \frac{\frac{x}{3} + 1}{4 x ^{2} + 4 x + 3} d x

Apply u-substitution

: \int \frac{3 u}{36 u ^{2} - 60 u + 27} d u

= 3 \cdot \int \frac{3 u}{36 u ^{2} - 60 u + 27} d u

Take the constant out:

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

= 3 \cdot 3 \cdot \int \frac{u}{36 u ^{2} - 60 u + 27} d u

Complete the square

36 u^{2} - 60 u + 27 : 36 (u - \frac{5}{6})^{2} + 2

= 3 \cdot 3 \cdot \int \frac{u}{36 ( u - \frac{5}{6} ) ^{2} + 2} d u

Apply u-substitution

: \int \frac{6 v + 5}{6 36 v ^{2} + 2} d v

= 3 \cdot 3 \cdot \int \frac{6 v + 5}{6 36 v ^{2} + 2} d v

Take the constant out:

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

= 3 \cdot 3 \cdot \frac{1}{6} \cdot \int \frac{6 v + 5}{36 v ^{2} + 2} d v

Expand

\frac{6 v + 5}{36 v ^{2} + 2} : \frac{6 v}{36 v ^{2} + 2} + \frac{5}{36 v ^{2} + 2}

= 3 \cdot 3 \cdot \frac{1}{6} \cdot \int \frac{6 v}{36 v ^{2} + 2} + \frac{5}{36 v ^{2} + 2} 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

= 3 \cdot 3 \cdot \frac{1}{6} (\int \frac{6 v}{36 v ^{2} + 2} d v + \int \frac{5}{36 v ^{2} + 2} d v)

\int \frac{6 v}{36 v ^{2} + 2} d v = \frac{1}{6} 36 v^{2} + 2

\int \frac{5}{36 v ^{2} + 2} d v = \frac{5}{6} ln 32 v + 1 + 18 v^{2}

= 3 \cdot 3 \cdot \frac{1}{6} (\frac{1}{6} 36 v^{2} + 2 + \frac{5}{6} ln 32 v + 1 + 18 v^{2})

Substitute back

= 3 \cdot 3 \cdot \frac{1}{6} \frac{1}{6} 36 (\frac{x}{3} + 1 - \frac{5}{6})^{2} + 2 + \frac{5}{6} ln 32 (\frac{x}{3} + 1 - \frac{5}{6}) + 1 + 18 (\frac{x}{3} + 1 - \frac{5}{6})^{2}

Simplify

3 \cdot 3 \cdot \frac{1}{6} \frac{1}{6} 36 (\frac{x}{3} + 1 - \frac{5}{6})^{2} + 2 + \frac{5}{6} ln 32 (\frac{x}{3} + 1 - \frac{5}{6}) + 1 + 18 (\frac{x}{3} + 1 - \frac{5}{6})^{2} : \frac{1}{4} (5 (ln 2 x + 2 \frac{4 x ^{2} + 4 x + 3}{2} + 1 - \frac{ln ( 2 )}{2}) + 4 x^{2} + 4 x + 3)

= \frac{1}{4} (5 (ln 2 x + 2 \frac{4 x ^{2} + 4 x + 3}{2} + 1 - \frac{ln ( 2 )}{2}) + 4 x^{2} + 4 x + 3)

Add a constant to the solution

= \frac{1}{4} (5 (ln 2 x + 2 \frac{4 x ^{2} + 4 x + 3}{2} + 1 - \frac{ln ( 2 )}{2}) + 4 x^{2} + 4 x + 3) + C

Popular Examples

integral of (27x^3-1)/(\sqrt[3]{x)}

integral of (e^{bt})/((a-e^{bt))^5}

\int \frac{e ^{b t}}{( a - e ^{b t} ) ^{5}} d t

integral of 5/((x-6)(x-1))

\int \frac{5}{( x - 6 ) ( x - 1 )} d x

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

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

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

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