Integration Problem

Calculus Level 2

∫ 0 2 x 2 x + 3 d x \large \int_0^2 x \sqrt{2x+3} \, dx

The value of the integral above is equal to 1 A ( B B + C C ) \dfrac1A (B\sqrt B + C \sqrt C ) , where A , B A,B and C C are positive integers with B , C B,C square-free. Find A + B + C A+B+C .


The answer is 15.

Stefanos Charkoutsis by stefanos charkoutsis , 24, United Kingdom

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1 solution

∫ 0 2 x 2 x + 3 d x = 1 4 ( 2 5 ( 2 x + 3 ) 5 2 − 2 ( 2 x + 3 ) 3 2 ) + C Indifinite = 1 5 7 7 − ( − 1 5 3 3 ) = 1 5 ( 3 3 + 7 7 ) \begin{aligned} \int_{0}^{2} x \sqrt{2x+3}\,dx &= \frac{1}{4}\left(\frac{2}{5}(2x+3)^{\frac{5}{2}} -2(2x+3)^{\frac{3}{2}}\right) + C \quad\quad\quad\quad{\text{Indifinite}} \\&= \frac{1}{5}7\sqrt{7} - \left(-\frac{1}{5}3\sqrt{3}\right) \\&= \frac{1}{5}(3\sqrt{3}+7\sqrt{7}) \end{aligned}

Were A + B + C = 15 □ A+B+C = 15 \space \square

ADIOS!!! \LARGE \text{ADIOS!!!}

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