Squares of Trigonometric Ratios (2)

Calculus Level 4

∫ 0 π / 2 d x 16 sin ⁡ 2 x + 9 cos ⁡ 2 x = ? \large \displaystyle \int_{0}^{\pi/2}\dfrac{dx}{16 \sin^2x + 9 \cos^2 x}= \, ?

Give your answer to 3 decimal places.

For your final step, you may use the approximation π = 22 7 \pi = \dfrac{22}{7} .


The answer is 0.13095.

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Samara Simha Reddy by Samara Simha Reddy , 22, India

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1 solution

∫ 0 π / 2 d x 16 sin ⁡ 2 x + 9 cos ⁡ 2 x = ∫ 0 π / 2 sec ⁡ 2 x . d x 16 tan ⁡ 2 x + 9 = 1 16 ∫ 0 π / 2 sec ⁡ 2 x . d x 9 16 + tan ⁡ 2 x = 1 16 × 4 3 [ tan ⁡ − 1 ( 4 tan ⁡ x 3 ) ] 0 π / 2 = 1 12 [ π 2 − 0 ] = π 24 = 22 24 × 7 = 22 168 = 0.13095 \begin{aligned} \large \displaystyle \int_{0}^{\pi/2}\frac{dx}{16\sin^2x + 9\cos^2x}\large \displaystyle &= \int_0^{\pi/2} \frac{\sec^2x . dx}{16\tan^2x + 9}\\ \large \displaystyle &= \frac{1}{16} \int_0^{\pi/2} \frac{\sec^2x . dx}{\frac{9}{16} + \tan^2x}\\ \large \displaystyle &= \frac{1}{16} \times \frac{4}{3} \left[\tan^{-1} \left(\frac{4\tan x}{3}\right) \right]_0^{\pi/2}\\ \large \displaystyle &= \frac{1}{12} \left[\frac{\pi}{2} - 0 \right] = \frac{\pi}{24}\\ \large \displaystyle &= \frac{22}{24\times 7} = \frac{22}{168} = \boxed{0.13095} \end{aligned}

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@Samara Simha Reddy : Hi! Thanks for posting such a beautiful question and a great solution, keep up the good work! ⌣ ¨ \ddot\smile

But while using latex for trigonometry please avoid typing "sin" simply, instead put a backslash to it like "\sin" and it will look better(like I have now edited your solution). Also, you can use the "\begin{align} . . . \end{align}" to align your solutions which will enhance the quality. For more information on latex check this: Latex Guide .

P.S: If you need more help or assistance, feel free to reply to this comment, or join your fellow Brilliant-ians in the slack website here .

Sravanth C. - 5 years, 2 months ago

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And how did you edit my solution??

Samara Simha Reddy - 5 years, 2 months ago

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Actually I am a this , and being that I have the powers ;)

Sravanth C. - 5 years, 2 months ago

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@Sravanth C. – How to be one among them??

Samara Simha Reddy - 5 years, 2 months ago

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@Samara Simha Reddy – It comes with hard work, if you have a great impact on the community, the staff or the moderators themselves can choose you. Well, you need to be very active to get the responsibility of being a moderator. Thanks!

Sravanth C. - 5 years, 2 months ago

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