What is the integral of xsqrt(x+7) ?

The integral of xsqrt(x+7) is 2/5 (x+7)^{5/2}-14/3 (x+7)^{3/2}+C

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integral of xsqrt(x+7)

\int x x + 7 d x

\frac{2}{5} (x + 7)^{\frac{5}{2}} - \frac{14}{3} (x + 7)^{\frac{3}{2}} + C

Solution steps

\int x x + 7 d x

Apply u-substitution

: \int (u - 7) u d u

= \int (u - 7) u d u

Expand

(u - 7) u : u^{\frac{3}{2}} - 7 u

= \int u^{\frac{3}{2}} - 7 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

= \int u^{\frac{3}{2}} d u - \int 7 u d u

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

\int 7 u d u = \frac{14}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} - \frac{14}{3} u^{\frac{3}{2}}

Substitute back

u = x + 7

= \frac{2}{5} (x + 7)^{\frac{5}{2}} - \frac{14}{3} (x + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (x + 7)^{\frac{5}{2}} - \frac{14}{3} (x + 7)^{\frac{3}{2}} + C

x \to - 2 + lim (x)

\frac{d}{d x} (\frac{sin ( 2 x + 1 )}{2 x + 1})

\int - x^{- 4} d x

\int t^{4} d t

y = x + 1

What is the integral of xsqrt(x+7) ?
The integral of xsqrt(x+7) is 2/5 (x+7)^{5/2}-14/3 (x+7)^{3/2}+C