First slide

Evaluation of definite integrals

Question

Let $f (x) = \sin x + \sum_{r = 0}^{2} \cos (x + \frac{2 r π}{3})$
$g (x) = \cos x + \sum_{r = 0}^{2} \sin (x + \frac{2 r π}{3})$
$h (x) = max {f (x), g (x)}$

NA

Question

$\int_{- π}^{π} g {f (x)} \sin 2 x dx$ is equalent to

A

$2 \int_{0}^{π} g {f (x)} \sin 2 x dx$
B

$4 \int_{0}^{π / 2} g {f (x)} \sin 2 x dx$
C

$\int_{- π / 2}^{π / 2} \cos^{2} x \cdot \sin^{5} xdx$
D

1

Solution

$\begin{array}{l} f (x) = \sin x + \sum_{r = 0}^{2} \cos (x + \frac{2 rπ}{3}) \\ = \sin x + \cos x + \cos (x + \frac{2 π}{3}) + \cos (x + \frac{4 π}{3}) \end{array}$
$= \sin x + \cos x - 2 \cdot \frac{1}{2} \cos x = \sin x$
similarly g(x) = cosx
$\int_{- π}^{π} g {f (x)} \sin 2 x dx = \int_{- π}^{π} \cos (sinx) \cdot \sin 2 x dx = 0$
since integrand is odd

Question

$\int_{π / 4}^{π} [h (x)] {f (x)}^{2} dx$ where [.] denotes greatest integer function is

A

$π$
B

$\frac{3 π}{4}$
C

0
D

$\frac{5 π}{4}$

Solution

$\int_{π / 4}^{π} [h (x)] {f (x)}^{2} dx = \int_{x / 4}^{π} [\sin x] \cdot \sin^{2} xdx = 0$

Question

$\int_{0}^{π} {g (x)}^{337} dx =$

A

0
B

$π$
C

$\frac{337 π}{2}$
D

1

Solution

$\begin{aligned} \int_{0}^{π} {g (x)}^{337} dx = \int_{0}^{π} \cos^{337} xdx = \int_{0}^{π} \cos^{337} (π - x) dx \\ = - \int_{0}^{π} \cos^{337} xdx ∴ \int_{0}^{π} \cos^{337} xdx = 0 \end{aligned}$

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