JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The integral $16 \int_{1}^{2} \frac{dx}{x^3 (x^2 + 2)^2}$ is equal to

Choose the correct answer:

Explanation

Solving

1. Substitution

Let $x^2 = t$ . Differentiating both sides:

2x dx = dt \implies dx = \frac{dt}{2x}

Substituting this into the integral:

I = 16 \int \frac{1}{x^3 (t + 2)^2} \cdot \frac{dt}{2x} = 8 \int \frac{dt}{x^4 (t + 2)^2}

Since $x^2 = t$ , then $x^4 = t^2$ :

I = 8 \int \frac{dt}{t^2 (t + 2)^2}

2. Change of Limits

When $x = 1$ , $t = 1^2 = 1$ .
When $x = 2$ , $t = 2^2 = 4$ .

The new integral is:

$I = 8 \int_{1}^{4} \frac{dt}{t^2 (t + 2)^2}$

3. Partial Fractions

We can decompose the integrand $\frac{1}{t^2 (t + 2)^2}$ :

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

\frac{1}{t^2 (t + 2)^2} = \frac{1}{4} \left[ \frac{1}{t^2} + \frac{1}{(t + 2)^2} - \frac{2}{t(t + 2)} \right]

Using partial fractions again for $\frac{2}{t(t + 2)} = \frac{1}{t} - \frac{1}{t + 2}$ :

\frac{1}{t^2 (t + 2)^2} = \frac{1}{4} \left[ \frac{1}{t^2} + \frac{1}{(t + 2)^2} - \left( \frac{1}{t} - \frac{1}{t + 2} \right) \right]

4. Integration

I = 8 \cdot \frac{1}{4} \int_{1}^{4} \left[ \frac{1}{t^2} + \frac{1}{(t + 2)^2} - \frac{1}{t} + \frac{1}{t + 2} \right] dt

I = 2 \left[ -\frac{1}{t} - \frac{1}{t + 2} - \log_{e} t + \log_{e}(t + 2) \right]_{1}^{4}

I = 2 \left[ \log_{e} \left( \frac{t + 2}{t} \right) - \frac{1}{t} - \frac{1}{t + 2} \right]_{1}^{4}

5. Evaluating at Limits

I = 2 \left[ \left( \log_{e} \frac{6}{4} - \frac{1}{4} - \frac{1}{6} \right) - \left( \log_{e} \frac{3}{1} - \frac{1}{1} - \frac{1}{3} \right) \right]

I = 2 \left[ \log_{e} \frac{3}{2} - \frac{5}{12} - \log_{e} 3 + \frac{4}{3} \right]

I = 2 \left[ (\log_{e} 3 - \log_{e} 2) - \log_{e} 3 + \frac{16 - 5}{12} \right]

I = 2 \left[ \frac{11}{12} - \log_{e} 2 \right]

I = \frac{11}{6} - 2 \log_{e} 2 = \frac{11}{6} - \log_{e} 4

Correct Option: (2)

Choose the correct answer:

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