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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate: $\int \limits_0^{\frac{z}{2}} \sin ^4 x \cos ^6 x dx$

\[\begin {array} {1 1} (a)\;\frac{5z}{256} \\ (b)\;\frac{3z}{512} \\ (c)\;\frac{3z}{256} \\ (d)\;\frac{3z}{1024} \end {array}\]
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1 Answer

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$m=4,n=6,k= \large\frac{z}{2}$
$\int \limits_0^{z/2} \sin ^m x\; \cos ^n x \;dx= \large\frac{\{(m-1)(m-3)....\}\{(n-1)(n-3)\}}{(m+n)(m+n-2)..}$$.k$
$m,n \neq 1\qquad m,n \in even => k= z/2 ;\qquad m,n \in odd => k= 1$
=> $\large\frac{(4-1)(4-3)(6-1)(6-3)(6-5)}{(10)(10-2)(10-4)(10-6)(10-8)} \frac{z}{2}$
=> $\large\frac{3 \times 1 \times 5 \times 3 \times 1}{10 \times 8 \times 6 \times 4 \times 2}$
=> $\large\frac{3z}{512}$
answered Dec 12, 2013 by meena.p
 
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