Search any question & find its solution
Question:
Answered & Verified by Expert
Integrate the function
$x \sqrt{x+2}$
$x \sqrt{x+2}$
Solution:
2727 Upvotes
Verified Answer
Let $x+2=t^2 \Rightarrow d x=2 t d t$
$\begin{aligned}
&\therefore \quad \int x \sqrt{x+2} d x=\int\left(t^2-2\right) t(2 t) d t \\
&=2 \int\left(t^4-2 t^2\right) d t=2 \frac{t^5}{5}-4 \frac{t^3}{3}+\mathrm{C} \\
&=\frac{2}{5}(x+2)^{5 / 2}-\frac{4}{3}(x+2)^{3 / 2}+\mathrm{C}
\end{aligned}$
$\begin{aligned}
&\therefore \quad \int x \sqrt{x+2} d x=\int\left(t^2-2\right) t(2 t) d t \\
&=2 \int\left(t^4-2 t^2\right) d t=2 \frac{t^5}{5}-4 \frac{t^3}{3}+\mathrm{C} \\
&=\frac{2}{5}(x+2)^{5 / 2}-\frac{4}{3}(x+2)^{3 / 2}+\mathrm{C}
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.