First slide

Evaluation of definite integrals

Question

If $f (0) = 2, f^{'} (x) = f (x), ϕ (x) = x + f (x)$ then $\int_{0}^{1} f (x) ϕ (x) d x$ is

Moderate

A

$e^{2}$
B

$2 e^{2}$
C

$2 e$
D

$2 e^{3 / 2}$

Solution

Since $f^{'} (x) = f (x)$ and $f (0) = 2$ so $f (x) = 2 e^{x},$ thus $ϕ (x) = x + 2 e^{x} .$ Hence
$\begin{array}{l} \int_{0}^{1} f (x) ϕ (x) d x = 2 \int_{0}^{1} (x e^{x} + 2 e^{2 x}) d x \\ = 2 [x e^{x} - {e^{x}|}_{0}^{1} + {e^{2 x}|}_{0}^{1}] \\ = 2 [([e - e] + 1) + e^{2} - 1] = 2 e^{2} . \end{array}$

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