Is Walli's Applicable?

Calculus Level 4

∫ 0 1 sin ⁡ 6 ( x ) d x = ? \large \int_0^1 \sin^6(x) \, dx = \ ?


The answer is 0.065.

Shashank Rustagi by Shashank Rustagi , 24, India

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1 solution

Noel Lo
Jun 18, 2015

express s i n 6 ( x ) = ( s i n 2 ( x ) ) 3 = ( 1 − c o s 2 x 2 ) 3 sin^6 (x) = (sin^2 (x))^3 = (\frac{1-cos 2x}{2})^3 .

For c o s 2 ( 2 x ) cos^2 (2x) , use double angle identity c o s 2 ( 2 x ) = 1 + c o s 4 x 2 cos^2 (2x) = \frac{1+cos 4x}{2} while for c o s 3 ( 2 x ) cos^3 (2x) , use triple angle identity c o s 6 x = 4 c o s 3 ( 2 x ) − 3 c o s ( 2 x ) cos 6x = 4cos^3 (2x) - 3cos (2x) .

Upon simplification, your final answer should be 5 16 − 15 64 s i n 2 + 3 64 s i n 4 − s i n 6 192 \frac{5}{16} - \frac{15}{64} sin 2 + \frac{3}{64} sin 4 - \frac{sin 6}{192} in exact form.

8 Helpful 0 Interesting 0 Brilliant 0 Confused

@Noel Lo I too did it the same way, but it was way too tedious.....I was wondering, do you know any shortcuts for this integral??

Aaghaz Mahajan - 3 years, 3 months ago

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