What is the integral of 3xsqrt(x+2) ?

The integral of 3xsqrt(x+2) is 6/5 (x+2)^{5/2}-4(x+2)^{3/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 3xsqrt(x+2)

\int 3 x x + 2 d x

\frac{6}{5} (x + 2)^{\frac{5}{2}} - 4 (x + 2)^{\frac{3}{2}} + C

Solution steps

\int 3 x x + 2 d x

Take the constant out:

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

= 3 \cdot \int x x + 2 d x

Apply u-substitution

: \int (u - 2) u d u

= 3 \cdot \int (u - 2) u d u

Expand

(u - 2) u : u^{\frac{3}{2}} - 2 u

= 3 \cdot \int u^{\frac{3}{2}} - 2 u d u

Apply the Sum Rule:

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

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

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

= 3 (\frac{2}{5} u^{\frac{5}{2}} - \frac{4}{3} u^{\frac{3}{2}})

Substitute back

u = x + 2

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

Simplify

3 (\frac{2}{5} (x + 2)^{\frac{5}{2}} - \frac{4}{3} (x + 2)^{\frac{3}{2}}) : \frac{6}{5} (x + 2)^{\frac{5}{2}} - 4 (x + 2)^{\frac{3}{2}}

= \frac{6}{5} (x + 2)^{\frac{5}{2}} - 4 (x + 2)^{\frac{3}{2}}

Add a constant to the solution

= \frac{6}{5} (x + 2)^{\frac{5}{2}} - 4 (x + 2)^{\frac{3}{2}} + C

\frac{\partial}{\partial y} (x^{4} y^{3} + 8 x^{2} y)

f^{^{'''}} + f^{^{''}} + 3 f^{^{'}} - 5 f = 0

s - 1

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

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

What is the integral of 3xsqrt(x+2) ?
The integral of 3xsqrt(x+2) is 6/5 (x+2)^{5/2}-4(x+2)^{3/2}+C