What is the integral of 5xsqrt(x+2) ?

The integral of 5xsqrt(x+2) is 2(x+2)^{5/2}-20/3 (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 5xsqrt(x+2)

\int 5 x x + 2 d x

2 (x + 2)^{\frac{5}{2}} - \frac{20}{3} (x + 2)^{\frac{3}{2}} + C

Solution steps

\int 5 x x + 2 d x

Take the constant out:

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

= 5 \cdot \int x x + 2 d x

Apply u-substitution

: \int (u - 2) u d u

= 5 \cdot \int (u - 2) u d u

Expand

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

= 5 \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

= 5 (\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}}

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

Substitute back

u = x + 2

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

Simplify

5 (\frac{2}{5} (x + 2)^{\frac{5}{2}} - \frac{4}{3} (x + 2)^{\frac{3}{2}}) : 2 (x + 2)^{\frac{5}{2}} - \frac{20}{3} (x + 2)^{\frac{3}{2}}

= 2 (x + 2)^{\frac{5}{2}} - \frac{20}{3} (x + 2)^{\frac{3}{2}}

Add a constant to the solution

= 2 (x + 2)^{\frac{5}{2}} - \frac{20}{3} (x + 2)^{\frac{3}{2}} + C

\int \frac{- 5}{3 x} d x

\int \frac{7}{( 2 x + 1 ) ( 1 + ln ( 2 x + 1 ) ) ^{2}} d x

\int 5 x^{2} (10 x^{3} - 7)^{5} d x

\int sin (x) (2 cos (x)) d x

\int csc^{5} (x) cot^{5} (x) d x

What is the integral of 5xsqrt(x+2) ?
The integral of 5xsqrt(x+2) is 2(x+2)^{5/2}-20/3 (x+2)^{3/2}+C