First slide

Evaluation of definite integrals

Question

The value of $\int_{1 / 2}^{1} (2 x \sin \frac{1}{x} - \cos \frac{1}{x}) d x$ is

Moderate

A

sin 1
B

cos 1
C

2 sin 1
D

none of these

Solution

$\begin{array}{l} \int_{1 / 2}^{1} [2 x \sin (\frac{1}{x}) - \cos (\frac{1}{x})] \\ {= x^{2} \sin (\frac{1}{x})]}_{1 / 2}^{1} - \int_{1 / 2}^{1} x^{2} (\cos (\frac{1}{x})) (- \frac{1}{x^{2}}) - \int_{1 / 2}^{1} \cos (\frac{1}{x}) d x \end{array}$

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