Question

The solution of \(\int_1^a {\int_1^b {\frac{{dxdy}}{{x\,y}}} } \) is
1. In(ab)
2. In(a/b)
3. In(a) + In(b)
4. In(a) In(b)

Answer 1

Correct Answer - Option 4 : In(a) In(b)

Concept:

Evaluation of Double integrals: It doesn’t matter which variable we integrate with respect to first, we will get the same answer regardless of the order of integration.

\(\int {{\frac{{dx}}{{x}}} } = ln (x) + c\)

Calculation:

Given:

I = \(\int_1^a {\int_1^b {\frac{{dxdy}}{{x\,y}}} } \)

= \(\int_1^a {{\frac{{dx}}{{x}}} × \int_1^b {\frac{{dy}}{{y}}} } \)

= | ln (x) |_{1 to a} × | ln (y) |_{1 to b} 

= [ln (a) - ln (1)] × [ln(b) - ln(1)]

I = ln (a) ln (b)