Which of the following statement is correct for the function g ( α ) for α ∈ R such that g α = ∫ π 6 π 3 sin α x cos α x + sin α x d x

g α = ∫ π 6 π 3 sin α x sin α x + cos α x d x … i g α = ∫ π 6 π 3 cos α x sin α x + cos α x d x … ii Adding 1 + 2 2 g α = π 3 - π 6 = π 6 ⇒ g α = π 12 Which is constant and an even function.

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Mathematics - Definite Integration Question with Solution | TestHub

MathematicsDefinite IntegrationProperties of definite integrationEasy2 minPYQ_2021

MathematicsEasysingle choice

Which of the following statement is correct for the function $g (α)$ for $α \in R$ such that $g (α) = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{\sin^{α} x}{\cos^{α} x + \sin^{α} x} d x$

Options:

Answer:

Solution:

$g (α) = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{\sin^{α} x}{(\sin^{α} x + \cos^{α} x)} d x \dots (i)$

$g (α) = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{\cos^{α} x}{(\sin^{α} x + \cos^{α} x)} d x \dots (ii)$

Adding $(1) + (2)$

$2 g (α) = \frac{π}{3} - \frac{π}{6} = \frac{π}{6}$

$\Rightarrow g (α) = \frac{π}{12}$

Which is constant and an even function.

Stream:JEESubject:MathematicsTopic:Definite IntegrationSubtopic:Properties of definite integration

⏱ 2mℹ️ Source: PYQ_2021

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