If $\int(7 x-2) \sqrt{3 x+2} \mathrm{~d} x=\mathrm{A}(3 x+2)^{\frac{5}{2}}+\mathrm{B}(3 x+2)^{\frac{3}{2}}+\mathrm{c}$ (where c is a constant of integration), then the values of $A$ and $B$ are respectively

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Aicharya · Accepted Answer

Let $\mathrm{I}=\int(7 x-2) \sqrt{3 x+2} \mathrm{~d} x$ Let $3 x+2=\mathrm{t} \Rightarrow x=\frac{\mathrm{t}-2}{3} \Rightarrow \mathrm{~d} x=\frac{1}{3} \mathrm{dt}$ $\begin{aligned} 	herefore \quad & \mathrm{I} \ & =\int\left[7\left(\frac{\mathrm{t}-2}{3}ight)-2ight] \sqrt{\mathrm{t}} \mathrm{dt} \ & =\frac{1}{3} \int\left(\frac{7 \mathrm{t}}{3}-\frac{20}{3}ight) \sqrt{\mathrm{t}} \mathrm{dt} \ & =\frac{7}{9} \int \mathrm{t}^{\frac{3}{2}} \mathrm{dt}-\frac{20}{9} \int \mathrm{t}^{\frac{1}{2}} \mathrm{dt} \ & =\frac{14}{45}(3 x+2)^{\frac{5}{2}}-\frac{40}{27}(3 x+2)^{\frac{3}{2}}+\mathrm{C} \ 	herefore \quad & A=\frac{14}{45} 	ext { and } B=\frac{-40}{27} \end{aligned}$

If $\int(7 x-2) \sqrt{3 x+2} \mathrm{~d} x=\mathrm{A}(3 x+2)^{\frac{5}{2}}+\mathrm{B}(3…

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