What is the integral of (x+5)sqrt(8-x) ?

The integral of (x+5)sqrt(8-x) is 2/5 (8-x)^{5/2}-26/3 (8-x)^{3/2}+C

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integral of (x+5)sqrt(8-x)

\int (x + 5) 8 - x d x

\frac{2}{5} (8 - x)^{\frac{5}{2}} - \frac{26}{3} (8 - x)^{\frac{3}{2}} + C

Solution steps

\int (x + 5) 8 - x d x

Apply u-substitution

: \int - u (- u + 13) d u

= \int - u (- u + 13) d u

Take the constant out:

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

= - \int u (- u + 13) d u

Expand

u (- u + 13) : - u^{\frac{3}{2}} + 13 u

= - \int - u^{\frac{3}{2}} + 13 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 13 u d u)

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

\int 13 u d u = \frac{26}{3} u^{\frac{3}{2}}

= - (- \frac{2}{5} u^{\frac{5}{2}} + \frac{26}{3} u^{\frac{3}{2}})

Substitute back

u = 8 - x

= - (- \frac{2}{5} (8 - x)^{\frac{5}{2}} + \frac{26}{3} (8 - x)^{\frac{3}{2}})

Simplify

= \frac{2}{5} (8 - x)^{\frac{5}{2}} - \frac{26}{3} (8 - x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (8 - x)^{\frac{5}{2}} - \frac{26}{3} (8 - x)^{\frac{3}{2}} + C

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

n = 0 \sum \infty (4 x^{2})^{n}

x \to π lim (9 sin (sin (x)))

\int \frac{4}{x + 2} d x

y^{^{''}} - 4 y^{^{'}} - 5 y = 0, y (0) = 3, y^{^{'}} (0) = 9

What is the integral of (x+5)sqrt(8-x) ?
The integral of (x+5)sqrt(8-x) is 2/5 (8-x)^{5/2}-26/3 (8-x)^{3/2}+C