\(\int_0^{\pi/4}\sin^3 \theta d\theta=?\)

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This question was previously asked in

AAI ATC Junior Executive 21 Feb 2023 Shift 1 Official Paper

Answer 1

दिया गया है:

\(\int_0^{\pi/4}\sin^3 \theta d\theta\)

अवधारणा:

सूत्र \(\rm sin^2\theta=1-cos^2\theta\) उपयोग कीजिए।

गणना:

\(\int_0^{\pi/4}\sin^3 \theta d\theta\)

\(\rm =\int_0^{\pi/4}(1-cos^2\theta)sin\theta \ d\theta\)

\(\rm =\int_0^{\pi/4}sin\theta \ d\theta-\int_0^{\pi/4}cos^2\theta\ sin\theta\ d \theta\)

\(\rm cos\theta=u \implies sin\theta=-d\theta\)

\(\rm =\int_0^{\pi/4}sin\theta \ d\theta+\int_0^{\pi/4}u^2\ du\)

\(\rm =[-cos\theta]_0^{\pi/4}+[\frac{cos^{3} \theta}{3}]_0^{\pi/4}\)

\(\rm =-[\frac{1}{\sqrt2}-1]+\frac{1}{3}[\frac{1}{2\sqrt2}-1]\)

\(\rm =\frac{2}{3}-\frac{5}{6\sqrt2}\)

अतः विकल्प (2) सही है।

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