Correct Answer - Option 3 : 4
Concept:
Integration by parts: Integration by parts is a method to find integrals of products
- The formula for integrating by parts is given by,
- ∫u v dx = u∫v dx −∫u' (∫v dx) dx
Where u is the function u(x) and v is the function v(x)
ILATE Rule: Usually, the preference order of this rule is based on some functions such as Inverse, Logarithm, Algebraic, Trigonometric and Exponent.
Calculation:
Let , I = \(\mathop \smallint \ln x\cdot1\;dx\)
Using integration by parts,
\(\begin{array}{l} = \quad \ln x \int d x-\int\left(\frac{1}{x} \cdot \int d x\right) d x \\= \quad x \cdot \ln x-\int d x \\ \end{array}\)
= x ln x - x
= x(ln x - 1)
Now,
\(\mathop \smallint \nolimits_1^2 \ln x\;dx\)
\(\begin{aligned} &=[x(\ln x-1)]_{1}^{2}\\ &=[2 \ln 2-2-(\ln 1-1)]\\ &=\left[\operatorname{ln}\left(2)^{2}-2-\ln 1+1\right]\right.\\ &=[\ln 4-1]\\ &=[\ln 4-\ln e]\\ &=\ln\frac{4}{e} \end{aligned}\)
= [2 ln 2 - 2 - (ln 1 - 1)]
= [ln (2)2 - 2 - ln 1 + 1]
= ln 4 - 1
= ln 4 - ln e (∵ ln e = 1)
= ln (4/e) = ln (A/e)
∴ The value of A is 4
Hence, option (3) is correct.
