First slide

Introduction to integration

Question

$\int \sqrt{x} e^{\sqrt{x}} dx$ is equal to

Moderate

A

$e^{\sqrt{x}} + C$
B

$2 {xe}^{\sqrt{x}} + C$
C

$(2 x - 4 \sqrt{x} + 4) e^{\sqrt{x}} + C$
D

None of these

Solution

Let, $= \int \sqrt{x} e^{\sqrt{x}} dx$
Put $\sqrt{x} = t \Rightarrow \frac{1}{2 \sqrt{x}} dx = dt$
$\begin{array}{l} ∴ I = 2 \int t^{2} e^{t} dt = 2 \{t^{2} e^{t} - 2 \int {te}^{t} dt\} \\ = 2 \{t^{2} e^{t} - 2 ({te}^{t} - e^{t})\} + C \\ = 2 \{t^{2} e^{t} - 2 {te}^{t} + 2 e^{t}\} + C \\ = (2 x - 4 \sqrt{x} + 4) e^{\sqrt{x}} + C \end{array}$

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