What is the integral of xsqrt(x-21) ?

The integral of xsqrt(x-21) is 2/5 (x-21)^{5/2}+14(x-21)^{3/2}+C

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integral of xsqrt(x-21)

\int x x - 21 d x

\frac{2}{5} (x - 21)^{\frac{5}{2}} + 14 (x - 21)^{\frac{3}{2}} + C

Solution steps

\int x x - 21 d x

Apply u-substitution

: \int (u + 21) u d u

= \int (u + 21) u d u

Expand

(u + 21) u : u^{\frac{3}{2}} + 21 u

= \int u^{\frac{3}{2}} + 21 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 21 u d u

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

\int 21 u d u = 14 u^{\frac{3}{2}}

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

Substitute back

u = x - 21

= \frac{2}{5} (x - 21)^{\frac{5}{2}} + 14 (x - 21)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} (x - 21)^{\frac{5}{2}} + 14 (x - 21)^{\frac{3}{2}} + C

f (x) = \frac{7 x - 4}{- 3 x + 3}

\int \frac{5}{x ^{2} + 18 x + 82} d x

\int_{0}^{13} \frac{1}{x ^{\frac{1}{4}}} d x

n \to \infty lim (a^{n})

y^{^{'}} = x y^{2} + 2 y^{2}

What is the integral of xsqrt(x-21) ?
The integral of xsqrt(x-21) is 2/5 (x-21)^{5/2}+14(x-21)^{3/2}+C