What is the integral of 2x(4x^2+3)^{3/2} ?

The integral of 2x(4x^2+3)^{3/2} is 1/10 (4x^2+3)^{5/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 2x(4x^2+3)^{3/2}

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

\frac{1}{10} (4 x^{2} + 3)^{\frac{5}{2}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \int \frac{( 4 u + 3 ) ^{\frac{3}{2}}}{2} d u

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

Apply u-substitution

: \int \frac{v ^{\frac{3}{2}}}{4} d v

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

Apply the Power Rule

: \frac{2}{5} v^{\frac{5}{2}}

= 2 \cdot \frac{1}{2} \cdot \frac{1}{4} \cdot \frac{2}{5} v^{\frac{5}{2}}

Substitute back

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

Simplify

2 \cdot \frac{1}{2} \cdot \frac{1}{4} \cdot \frac{2}{5} (4 x^{2} + 3)^{\frac{5}{2}} : \frac{1}{10} (4 x^{2} + 3)^{\frac{5}{2}}

= \frac{1}{10} (4 x^{2} + 3)^{\frac{5}{2}}

Add a constant to the solution

= \frac{1}{10} (4 x^{2} + 3)^{\frac{5}{2}} + C

\int \frac{4 x ^{2} + 16 x + 16}{x ^{3} + 8 x ^{2} + 16 x} d x

\int \frac{x}{( 5 - 4 x ) ^{\frac{3}{2}}} d x

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

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

\int \frac{18}{x ^{3} + x ^{2}} d x

What is the integral of 2x(4x^2+3)^{3/2} ?
The integral of 2x(4x^2+3)^{3/2} is 1/10 (4x^2+3)^{5/2}+C