First slide

Evaluation of definite integrals

Question

$Value of \int_{\frac{3 π}{4}}^{π} [\sin x + [\frac{4 x}{π}]] d x where [] denotes step function is$

Moderate

A

0
B

$\frac{5 π}{4}$
C

$\frac{7 π}{4}$
D

$\frac{3 π}{4}$

Solution

We have $\frac{3 π}{4} < x < π \Rightarrow 3 < \frac{4 x}{π} < 4$
$\begin{array}{l} ∴ \int_{\frac{3 π}{4}}^{π} [\sin x + [\frac{4 x}{π}]] d x = \int_{\frac{3 π}{4}}^{π} [\sin x + 3] d x \\ = \int_{\frac{3 π}{4}}^{π} (3 + [\sin x]) d x \\ = 3 (π - \frac{3 π}{4}) + 0 (∵ [\sin x] = 0 f o r \frac{3 π}{4} < x < π) \\ = \frac{3 π}{4} \end{array}$

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