Mathematics

$$\int _{ 0 }^{ \pi /4 }{ \sec ^{ 7 }{ \theta  } \sin ^{ 3 }{ \theta  } d\theta  } $$


ANSWER

$$5/12$$


SOLUTION
Let $$\displaystyle I=\int_{0}^{\dfrac{\pi }{4}} \sec^{7}\theta \sin^{3}\theta d\theta $$
$$=\displaystyle  \int_{0}^{\dfrac{\pi }{4}}\sec^{4}\theta  \tan^{3}\theta d\theta $$
$$=\displaystyle \int_{0}^{\dfrac{\pi }{4}}\sec^{2}\theta \tan^{5}\theta +\int_{0}^{\dfrac{\pi }{4}}\tan^{3}\theta \sec^{2}\theta d\theta $$
Let, $$\tan\theta =4, du=\sec^{2}\theta d\theta $$
$$\theta \epsilon \left(0,\dfrac{\pi }{4}\right),4\epsilon (0,1)$$
$$\therefore \displaystyle I=\int_{0}^{1}x^{5}du+\int_{0}^{1}u^{3}du$$
$$I=\displaystyle \frac{1}{6}+\frac{1}{4}=\frac{5}{12}$$
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Single Correct Medium Published on 17th 09, 2020
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