What is the integral from 0 to 24 of xsqrt(x+a) ?

The integral from 0 to 24 of xsqrt(x+a) is (2(24+a)^{5/2}-2a^{5/2})/5-(a(2(24+a)^{3/2}-2a^{3/2}))/3

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integral from 0 to 24 of xsqrt(x+a)

\int_{0}^{24} x x + a d x

\frac{2 ( 24 + a ) ^{\frac{5}{2}} - 2 a ^{\frac{5}{2}}}{5} - \frac{a ( 2 ( 24 + a ) ^{\frac{3}{2}} - 2 a ^{\frac{3}{2}} )}{3}

Solution steps

\int_{0}^{24} x x + a d x

Apply u-substitution

: \int_{a}^{24 + a} (u - a) u d u

= \int_{a}^{24 + a} (u - a) u d u

Expand

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

= \int_{a}^{24 + a} u^{\frac{3}{2}} - a 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_{a}^{24 + a} u^{\frac{3}{2}} d u - \int_{a}^{24 + a} a u d u

\int_{a}^{24 + a} u^{\frac{3}{2}} d u = \frac{2 ( 24 + a ) ^{\frac{5}{2}} - 2 a ^{\frac{5}{2}}}{5}

\int_{a}^{24 + a} a u d u = \frac{a ( 2 ( 24 + a ) ^{\frac{3}{2}} - 2 a ^{\frac{3}{2}} )}{3}

= \frac{2 ( 24 + a ) ^{\frac{5}{2}} - 2 a ^{\frac{5}{2}}}{5} - \frac{a ( 2 ( 24 + a ) ^{\frac{3}{2}} - 2 a ^{\frac{3}{2}} )}{3}

\int_{0}^{1} \frac{sin ( x )}{x ^{2}} d x

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\int_{1}^{4} \frac{5}{t} d t

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\int_{1}^{4} 4 - \frac{4}{x} d x

What is the integral from 0 to 24 of xsqrt(x+a) ?
The integral from 0 to 24 of xsqrt(x+a) is (2(24+a)^{5/2}-2a^{5/2})/5-(a(2(24+a)^{3/2}-2a^{3/2}))/3