First slide

Evaluation of definite integrals

Question

$The value of the definite integral$ $\int_{0}^{3 π / 4} [(1 + x) \sin x + (1 - x) \cos x] d x is -$

Moderate

A

$2 \tan \frac{3 π}{8}$
B

$2 \tan \frac{π}{4}$
C

$2 \tan \frac{π}{8}$
D

0

Solution

$I = \int_{0}^{3 π / 4} (sinx + cosx) dx + \int_{0}^{3 π / 4} \underset{I}{\underset{⏟}{x}} (\underset{II}{\underset{⏟}{sinx - cosx}}) dx$
$= \int_{0}^{3 π / 4} (\sin x + \cos x) d x + \underset{zero}{\underset{⏟}{{x (- \cos x - \sin x)|}_{0}^{3 π / 4}}}$ $+ \int_{0}^{3 π / 4} (\sin x + \cos x) d x$
$= 2 \int_{0}^{3 π / 4} (\sin x + \cos x) d x = 2 (\sqrt{2} + 1) = 2 \tan \frac{3 π}{8}$

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