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

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

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integral of (5x)/(sqrt(x+2))

\int \frac{5 x}{x + 2} d x

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

Solution steps

\int \frac{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 \frac{x}{x + 2} d x

Apply u-substitution

= 5 \cdot \int 2 (u^{2} - 2) d u

Take the constant out:

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

= 5 \cdot 2 \cdot \int u^{2} - 2 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 \cdot 2 (\int u^{2} d u - \int 2 d u)

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

\int 2 d u = 2 u

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

Substitute back

u = x + 2

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

Simplify

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

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

Add a constant to the solution

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

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