<p><strong>254. <math><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mrow><mfrac bevelled="true"><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msubsup><mrow><mfrac><mrow><mi>c</mi><mi>o</mi><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi></mrow><mrow><mi>c</mi><mi>o</mi><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi><mo>+</mo><mn>4</mn><mi>s</mi><mi>i</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>x</mi></mrow></mfrac></mrow></mrow></mstyle><mi>d</mi><mi>x</mi></mrow></math></strong></p>

=∫0π2cos2xcos2x+4sin2xdx= ∫0π2cos2xcos2x+4(1−cos2x)dx {?1=cos2x+sin2x.= ∫0π2cos2xcos2x+4−4cos2xdx= ∫0π2cos2x4−3cos2xdx= −13∫0π2−3cos2x4−3cos2xdx= −13∫0π2(4−3cos2x)−44−3cos2xdx= −13[∫0π2dx−∫0π244−3cos2xdx]= −13{[x]0π2−∫0π244−3×1sec2xdx}= −13{π2−∫0π24sec2x4sec2x−3dx}= −13{π2−∫0π24sec2x4(1+tan2x)−3dx} {sen2x=1+tan2x}= −π6+231where I1 = ∫0π22tan2x1+4tan2xdx.Putting 2tan x = t =>2 sin2xdx = dt& when x = 0, t = 2 tan (0) = 0x = π2 , t = 2 tan π2 = ∞I1 = ∫0∞dt1+t2.= [tan−t]0∞= tan–1 (∞) – tan–10= π2−0 = π2.? I = −π6+23×π2=−π6+π3=π6.

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254. $\int_{0}^{\frac{π}{2}} \frac{c o s^{2} x}{c o s^{2} x + 4 s i n^{2} x} d x$

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Vishal Baghel | Contributor-Level 10

3 months ago

$= \int_{0}^{\frac{π}{2}} \frac{c o s^{2} x}{c o s^{2} x + 4 s i n^{2} x} d x$

= $\int_{0}^{\frac{π}{2}} \frac{c o s^{2} x}{c o s^{2} x + 4 (1 - c o s^{2} x)} d x {? 1 = c o s^{2} x + s i n^{2} x .$

= $\int_{0}^{\frac{π}{2}} \frac{c o s^{2} x}{c o s^{2} x + 4 - 4 c o s^{2} x} d x$

= $\int_{0}^{\frac{π}{2}} \frac{c o s^{2} x}{4 - 3 c o s^{2} x} d x$

= $- \frac{1}{3} \int_{0}^{\frac{π}{2}} \frac{- 3 c o s^{2} x}{4 - 3 c o s^{2} x} d x$

= $- \frac{1}{3} \int_{0}^{\frac{π}{2}} \frac{(4 - 3 c o s^{2} x) - 4}{4 - 3 c o s^{2} x} d x$

= $- \frac{1}{3} [\int_{0}^{\frac{π}{2}} d x - \int_{0}^{\frac{π}{2}} \frac{4}{4 - 3 c o s^{2} x} d x]$

= $- \frac{1}{3} {{[x]}_{0}^{\frac{π}{2}} - \int_{0}^{\frac{π}{2}} \frac{4}{4 - 3 \times \frac{1}{s e c^{2} x}} d x}$

= $- \frac{1}{3} {\frac{π}{2} - \int_{0}^{\frac{π}{2}} \frac{4 s e c^{2} x}{4 s e c^{2} x - 3} d x}$

= $- \frac{1}{3} {\frac{π}{2} - \int_{0}^{\frac{π}{2}} \frac{4 s e c^{2} x}{4 (1 + t a n^{2} x) - 3} d x} {s e n^{2} x = 1 + t a n^{2} x}$

= $\frac{- π}{6} + \frac{2}{3}_{1}$

where I₁ = $\int_{0}^{\frac{π}{2}} \frac{2 t a n^{2} x}{1 + 4 t a n^{2} x} d x .$

Putting 2tan x = t =>2 sin²xdx = dt& when x = 0, t = 2 tan (0) = 0

x = $\frac{π}{2}$ , t = 2 tan $\frac{π}{2}$ = ∞

I1 = $\int_{0}^{\infty} \frac{d t}{1 + t^{2}} .$

= $[t a n^{-} t]_{0}^{\infty}$

= tan^–1 (∞) – tan^–10

= $\frac{π}{2} - 0$ = $\frac{π}{2} .$

? I = $\frac{- π}{6} + \frac{2}{3} \times \frac{π}{2} = - \frac{π}{6} + \frac{π}{3} = \frac{π}{6} .$

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254. $\int_{0}^{\frac{π}{2}} \frac{c o s^{2} x}{c o s^{2} x + 4 s i n^{2} x} d x$