Elvaluate ∫ 0 π /2 1 + c o s 2 x s i n x d x

Let I = ∫ 0 π /2 1 + c o s 2 x s i n x dx Let cos x = t and − sin xdx = dt Now, x = 0 ⇒ t = cos 0 = 1 and x = 2 π ⇒ t = cos 2 π = 0 ∴ I = ∫ 1 0 1 + t 2 s i n x ( s i n x − dt ) = − ∫ 1 0 1 + t 2 dt = [ tan − 1 t ] 1 0 = − [ 0 − 4 π ] = 4 π

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MathematicsDefinite IntegrationDefinite Integration by SubstitutionMedium2 minPYQ_2013

MathematicsMediumsingle choice

Elvaluate $\int_{0}^{π /2} \frac{s i n x}{1 + c o s ^{2} x} d x$

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Let $I = \int_{0}^{π /2} \frac{s i n x}{1 + c o s ^{2} x} dx$ Let $cos x = t$ and $- sin xdx = dt$ Now, $x = 0 \Rightarrow t = cos 0 = 1$ and $x = \frac{π}{2} \Rightarrow t = cos \frac{π}{2} = 0$ $∴ I = \int_{1}^{0} \frac{s i n x}{1 + t ^{2}} (\frac{- dt}{s i n x}) = - \int_{1}^{0} \frac{dt}{1 + t ^{2}}$ $= [tan^{- 1} t]_{1}^{0} = - [0 - \frac{π}{4}] = \frac{π}{4}$

Stream:BITSATSubject:MathematicsTopic:Definite IntegrationSubtopic:Definite Integration by Substitution

⏱ 2mℹ️ Source: PYQ_2013

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