SATHEE: Indefinite Integration 1 Question 57

Indefinite Integration 1 Question 57

58. For $n > 0, \int_{0}^{2 π} \frac{x \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x =$

$(1997, 2 M)$

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Solution:

Let $I = \int_{0}^{2 π} \frac{x \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x$

$I = \int_{0}^{2 π} \frac{(2 π - x) [\sin (2 π - x)]^{2 n}}{[\sin (2 π - x)]^{2 n} + [\cos (2 π - x)]^{2 n}} d x$

$[∵ \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x]$

$I = \int_{0}^{2 π} \frac{(2 π - x) \cdot \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x$

$\Rightarrow I = \int_{0}^{2 π} \frac{2 π \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x - \int_{0}^{2 π} \frac{x \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x$

$\Rightarrow I = \int_{0}^{2 π} \frac{2 π \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x - I$

[from Eq. (i)]

$\Rightarrow I = \int_{0}^{2 π} \frac{π \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x$

$\Rightarrow I = π \int_{0}^{π} \frac{π \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x$

$+ \int_{0}^{π} \frac{\sin^{2 n} (2 π - x)}{\sin^{2 n} (2 π - x) + \cos^{2 n} (2 π - x)} d x$

$using property$

$\int_{0}^{2 a} f (x) d x = \int_{0}^{a} [f (x) + f (2 a - x)] d x$

$I = π \int_{0}^{π} \frac{\sin^{2 n} x d x}{\sin^{2 n} x + \cos^{2 n} x} d x$

$+ \int_{0}^{π} \frac{\sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x$

$\Rightarrow I = 2 π \int_{0}^{π} \frac{\sin^{2 n} x d x}{\sin^{2 n} x + \cos^{2 n} x} d x$

$\Rightarrow I = 4 π \int_{0}^{π / 2} \frac{\sin^{2 n} x d x}{\sin^{2 n} x + \cos^{2 n} x} d x$

$\Rightarrow I = 4 π \int_{0}^{π / 2} \frac{\sin^{2 n} (π / 2 - x)}{\sin^{2 n} (π / 2 - x) + \cos^{2 n} (π / 2 - x)} d x$

$\Rightarrow I = 4 π \int_{0}^{π / 2} \frac{\cos^{2 n} x}{\cos^{2 n} x + \sin^{2 n} x} d x$

On adding Eqs. (ii) and (iii), we get

$\begin{aligned} 2 I = 4 π \int_{0}^{π / 2} \frac{\sin^{2 n} x + \cos^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x \\ \Rightarrow 2 I & = 4 π \int_{0}^{π / 2} 1 d x = 4 π [x]_{0}^{π / 2} = 4 π \cdot \frac{π}{2} \\ \Rightarrow & I & = π^{2} \end{aligned}$

Indefinite Integration 1 Question 57

58. For $n > 0, \int_{0}^{2 π} \frac{x \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x =$

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Indefinite Integration 1 Question 57

58. For n>0,∫02πxsin2n⁡xsin2n⁡x+cos2n⁡xdx=

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58. For $n > 0, \int_{0}^{2 π} \frac{x \sin^{2 n} x}{\sin^{2 n} x + \cos^{2 n} x} d x =$