Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals2 Marks
Question
Integrate the function: $x \sqrt{x+2}$
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Answer
Let I = $ \int x \sqrt{x+2} d x$ Put (x + 2) = t $\Rightarrow$ dx = dt $\Rightarrow I = \int x \sqrt{x+2} d x=\int(t-2) \sqrt{t} d t$ $=\int\left(t^{\frac{3}{2}}-2 t^{\frac{t}{2}}\right) d t$ $=\int t^{\frac{3}{2}} d t-2 \int t^{\frac{1}{2}} d t$ $=\frac{t^{\frac{5}{2}}}{\frac{5}{2}}-2\left(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right)+C$ $=\frac{2}{5} t^{\frac{5}{2}}-\frac{4}{3} t^{\frac{3}{2}}+C$ $=\frac{2}{5}(x+2)^{\frac{5}{2}}-\frac{4}{3}(x+2)^{\frac{3}{2}}+C$
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