If f≤ft(x-1/x+1)=x+1, then f(x) d x=·s

Correct Answer is : −2log⁡∣1−x∣+c-2 |1-x|+c−2log∣1−x∣+c

Correct Answer is : −2log⁡∣1−x∣+c-2 |1-x|+c−2log∣1−x∣+c

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MHTCET 16 Apr 2026 Shift 2 Paper

Question : 118 of 150

Marks: +1, -0

f\left(\frac{x-1}{x+1}\right)=x+1

\int f(x) d x=\cdots

Solution:

Concept:

First find the explicit form of

f(x)

using a substitution, then integrate the resulting expression.

Explanation:

Let

y = \frac{x-1}{x+1}

Rearrange to express

x

in terms of

y

y(x+1) = x-1

yx + y = x - 1

yx - x = -1 - y

x(y-1) = -(1+y)

x = \frac{1+y}{1-y}

Now compute

x+1

x+1 = \frac{1+y}{1-y} + 1 = \frac{1+y+1-y}{1-y} = \frac{2}{1-y}

Hence,

f(y) = \frac{2}{1-y}

Therefore,

f(x) = \frac{2}{1-x}

Now integrate

f(x)

\int f(x)\,dx = \int \frac{2}{1-x}\,dx

Since

\frac{d}{dx}(1-x) = -1

, use the standard formula:

\int \frac{2}{1-x}\,dx = -2\log|1-x| + c

Answer:

-2\log|1-x| + c

, which corresponds to Option B.

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