1. ∫tan^3 (2x)sec^5 (2x) dx 2. ∫_0 ^(π/3) tan^5 (x)sec^6 (x) dx 3. ∫tan^6 (ay) dy


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Question Number 38130 by gunawan last updated on 22/Jun/18

$1 . \int \tan^{3} (2 x) \sec^{5} (2 x) d x$ $2 . \int_{0}^{\frac{π}{3}} \tan^{5} (x) \sec^{6} (x) d x$ $3 . \int \tan^{6} (a y) d y$
	Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jun/18
	$1)∫((sin^2 (2x)sin(2x))/(cos^8 (2x)))dx t=cos2x dt=−2sin2xdx =((−1)/2)∫((1−t^2 )/t^8 )×dt =((−1)/2){(t^(−7) /(−7))−(t^(−5) /((−5)))}+c (1/(14))×(1/(cos^7 (2x)))−(1/(10))×(1/(cos^5 (2x)))+c$
	$1) \int \frac{{s i n}^{2} (2 x) s i n (2 x)}{{c o s}^{8} (2 x)} d x$ $t = c o s 2 x d t = - 2 s i n 2 x d x$ $= \frac{- 1}{2} \int \frac{1 - t^{2}}{t^{8}} \times d t$ $= \frac{- 1}{2} {\frac{t^{- 7}}{- 7} - \frac{t^{- 5}}{(- 5)}} + c$ $\frac{1}{14} \times \frac{1}{{c o s}^{7} (2 x)} - \frac{1}{10} \times \frac{1}{{c o s}^{5} (2 x)} + c$
	Answered by Joel579 last updated on 22/Jun/18

	$(3)$ $I = \int \tan^{6} (a y) d y (u = a y \to d u = a d y)$ $= \frac{1}{a} \int {(\sec^{2} u - 1)}^{3} d u$ $= \frac{1}{a} \int (\sec^{6} u - {3 \sec}^{4} u + {3 \sec}^{2} u - 1) d u$
	Commented by Joel579 last updated on 22/Jun/18

	Answered by Joel579 last updated on 22/Jun/18

	$(3)$ $I = \int \tan^{6} (a y) d y (u = a y \to d u = a d y)$ $= \frac{1}{a} \int \tan^{2} u . \tan^{4} u d u$ $= \frac{1}{a} \int (\sec^{2} u . \tan^{4} u - \tan^{4} u) d u$ $= \frac{1}{a} \int [\sec^{2} u . \tan^{4} u - (\sec^{2} u - 1) \tan^{2} u] d u$ $= \frac{1}{a} \int [\sec^{2} u . \tan^{4} u - \sec^{2} u . \tan^{2} u] d u + \int \tan^{2} u d u$ $t = \tan u \to d t = \sec^{2} u d u$ $I = \frac{1}{a} \int t^{4} - t^{2} d t + \tan u - u$ $= \frac{1}{a} (\frac{t^{5}}{5} - \frac{t^{3}}{3} + \tan u - u) + C$
	Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jun/18
	$2)∫_0 ^(Π/3) ((sin^5 x)/(cos^5 x))×(1/(cos^6 x))dx =∫_0 ^(Π/3) ((sinx×(1−cos^2 x)^2 dx)/(cos^(11) x)) =(((−1))/)∫_0 ^(Π/3) ((1−2cos^2 x+cos^4 x)/(cos^(11) x))×d(cosx) =(−1)∫_0 ^(Π/3) {cos^(−11) x−2cos^(−9) x+cos^(−7) (x)}(dcosx) =∣((−1)/(−10))×cos^(−10) x+2((cos^(−8) x)/(−8))−(1/(−6))×cos^(−6) x∣_0 ^(Π/3)$
	$2) \int_{0}^{\frac{Π}{3}} \frac{{s i n}^{5} x}{{c o s}^{5} x} \times \frac{1}{{c o s}^{6} x} d x$ $= \int_{0}^{\frac{Π}{3}} \frac{s i n x \times {(1 - {c o s}^{2} x)}^{2} d x}{{c o s}^{11} x}$ $= \frac{(- 1)}{} \int_{0}^{\frac{Π}{3}} \frac{1 - 2 {c o s}^{2} x + {c o s}^{4} x}{{c o s}^{11} x} \times d (c o s x)$ $= (- 1) \int_{0}^{\frac{Π}{3}} {{c o s}^{- 11} x - 2 {c o s}^{- 9} x + {c o s}^{- 7} (x)} (d c o s x)$ $=∣ \frac{- 1}{- 10} \times {c o s}^{- 10} x + 2 \frac{{c o s}^{- 8} x}{- 8} - \frac{1}{- 6} \times {c o s}^{- 6} x ∣_{0}^{\frac{Π}{3}}$
	Commented by gunawan last updated on 22/Jun/18

	$Thank You very much Sir$
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