Let α and βα<β are roots of 18x2-9πx+π2=0, fx=x2, gx=cosx. Then ∫αβxgofxdx=

AP EAMCET · Maths · Definite Integration

Let $α$ and $β (α < β)$ are roots of $18 x^{2} - 9 π x + π^{2} = 0, f (x) = x^{2}, g (x) = \cos x$ . Then $\int_{α}^{β} x (g o f (x)) d x =$

A $\frac{\sqrt{3} - 1}{4}$
B $\frac{\sqrt{3}}{4}$
C $\frac{2 + \sqrt{3}}{2}$
D $\frac{1}{2} (\sin \frac{π^{2}}{9} - \sin \frac{π^{2}}{36})$

Verified Solution

Answer & Solution

Correct Answer

(D) $\frac{1}{2} (\sin \frac{π^{2}}{9} - \sin \frac{π^{2}}{36})$

Step-by-step Solution

Detailed explanation

Given fx=x2, gx=cosx i.e. gfx=cosx2 For 18x2-9πx+π2=0 roots are 9π±81π2-72π236=9π±3π36=π3, π6 i.e. α=π6, β=π3 So, ∫αβxg∘fxdx=∫π6π3xcosx2dx Let…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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