AP EAMCET · Maths · Definite Integration

$\int_{0}^{\frac{π}{4}} e^{\tan^{2} θ} \sin^{2} θ \tan θ d θ =$

A $\frac{1}{2} (\frac{e}{2} - 1)$
B $\frac{e}{2} - 1$
C $\frac{π}{2}$
D $2 (\frac{π}{2} - e)$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{1}{2} (\frac{e}{2} - 1)$

Step-by-step Solution

Detailed explanation

I=∫0π4etan2θsin2θtanθdθ =∫0π4etan2θtan3θsec2θ1+tan2θ2dθ Let tanθ=t I=∫01et2t31+t22dt Let t2=u I=∫01euu21+u2du=12∫01euu11+u2du By integrating by parts method…

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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∫0π4etan2θsin2θtanθdθ=

Step-by-step Solution

$\int_{0}^{\frac{π}{4}} e^{\tan^{2} θ} \sin^{2} θ \tan θ d θ =$