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Indefinite Integrals question

2005 · Shift 0 · Q78
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Indefinite Integrals question

2005 · Shift 0 · Q78

JEE MainMathematicsIndefinite IntegralsMCQ+4 / −1
∫{(log⁡x−1)1+(log⁡x)2}2  dx\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx}∫{1+(logx)2(logx−1)​}2dx is equal to
  1. A
    log⁡x(log⁡x)2+1+C{{\log x} \over {{{\left( {\log x} \right)}^2} + 1}} + C(logx)2+1logx​+C
  2. B
    xx2+1+C{x \over {{x^2} + 1}} + Cx2+1x​+C
  3. C
    xex1+x2+C{{x{e^x}} \over {1 + {x^2}}} + C1+x2xex​+C
  4. D
    x(log⁡x)2+1+C{x \over {{{\left( {\log x} \right)}^2} + 1}} + C(logx)2+1x​+C
View written solutionFree

Correct answer: D

  1. We need to evaluate I=∫(log⁡x−11+(log⁡x)2)2dx.I=\int \left(\frac{\log x-1}{1+(\log x)^2}\right)^2 dx.I=∫(1+(logx)2logx−1​)2dx.

  2. Since the options suggest expressions involving x1+(log⁡x)2\dfrac{x}{1+(\log x)^2}1+(logx)2x​, let us differentiate f(x)=x1+(log⁡x)2.f(x)=\frac{x}{1+(\log x)^2}.f(x)=1+(logx)2x​.

  3. Differentiate using the quotient rule: f′(x)=(1+(log⁡x)2)⋅ddx(x)−x⋅ddx(1+(log⁡x)2)(1+(log⁡x)2)2.f'(x)=\frac{(1+(\log x)^2)\cdot \frac{d}{dx}(x)-x\cdot \frac{d}{dx}(1+(\log x)^2)}{(1+(\log x)^2)^2}.f′(x)=(1+(logx)2)2(1+(logx)2)⋅dxd​(x)−x⋅dxd​(1+(logx)2)​. Now, ddx(x)=1,\frac{d}{dx}(x)=1,dxd​(x)=1, and ddx(1+(log⁡x)2)=2log⁡x⋅1x.\frac{d}{dx}(1+(\log x)^2)=2\log x\cdot \frac{1}{x}.dxd​(1+(logx)2)=2logx⋅x1​. So, f′(x)=1+(log⁡x)2−x(2log⁡x⋅1x)(1+(log⁡x)2)2.f'(x)=\frac{1+(\log x)^2-x\left(2\log x\cdot \frac{1}{x}\right)}{(1+(\log x)^2)^2}.f′(x)=(1+(logx)2)21+(logx)2−x(2logx⋅x1​)​. This simplifies to f′(x)=1+(log⁡x)2−2log⁡x(1+(log⁡x)2)2.f'(x)=\frac{1+(\log x)^2-2\log x}{(1+(\log x)^2)^2}.f′(x)=(1+(logx)2)21+(logx)2−2logx​.

  4. Rewrite the numerator: 1+(log⁡x)2−2log⁡x=(log⁡x−1)2.1+(\log x)^2-2\log x=(\log x-1)^2.1+(logx)2−2logx=(logx−1)2. Hence,

=\left(\frac{\log x-1}{1+(\log x)^2}\right)^2.$$ 5. Therefore, $$\int \left(\frac{\log x-1}{1+(\log x)^2}\right)^2 dx=\frac{x}{1+(\log x)^2}+C.$$ 6. Matching with the options, this is: $$\boxed{\text{D}}.$$ 7. Comparison with stored correct answer: - Stored correct answer: D - Derived answer: D So they agree.
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