₀^{π/4} { ² x}{(1+ x)(2+ x)} d x=

Correct Answer is : log⁡(43) ≤ft(4/3)log(34)

Correct Answer is : log⁡(43) ≤ft(4/3)log(34)

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MHTCET 15 May 2024 Shift 2 Paper

Question : 147 of 150

Marks: +1, -0

\int_{0}^{\frac{\pi}{4}} \frac{\sec ^{2} x}{(1+\tan x)(2+\tan x)} d x=

Solution:

Concept:

Use substitution

\tan x = t

to simplify the integrand into partial fractions.

Explanation:

Let

1+\tan x = t

Then

\sec^2 x \, dx = dt

Limits: when

x=0

t=1

; when

x=\frac{\pi}{4}

t=2

The integral becomes:

\int_{1}^{2} \frac{dt}{t(1+t)}

Split into partial fractions:

\frac{1}{t(1+t)} = \frac{1}{t} - \frac{1}{1+t}

So,

\int_{1}^{2} \left( \frac{1}{t} - \frac{1}{1+t} \right) dt = \left[ \log t - \log(1+t) \right]_{1}^{2}

Evaluate at limits:

= (\log 2 - \log 3) - (\log 1 - \log 2)

Since

\log 1 = 0

= \log 2 - \log 3 + \log 2 = \log 4 - \log 3 = \log \left( \frac{4}{3} \right)

Answer:

\log \left( \frac{4}{3} \right)

Hence, the correct option is C.

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