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STD 12 - 7.2 definite integral — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 7.2 definite integral4 Marks
MCQ
If $\frac{{d[f(x)]}}{{dx}} = g(x)$ for $a \le x \le b,$ then $\int_a^b {f(x)\,\,g(x)\,dx} $ equals
  • A
    $f(b) - f(a)$
  • B
    $g(b) - g(a)$
  • ✓
    $\frac{{{{[f(b)]}^2} - {{[f(a)]}^2}}}{2}$
  • D
    $\frac{{{{[g(b)]}^2} - {{[g(a)]}^2}}}{2}$
✓

Answer

Correct option: C.
$\frac{{{{[f(b)]}^2} - {{[f(a)]}^2}}}{2}$
c
(c) Let $I = \int_a^b {f(x)g(x)dx} $
Put $f(x) = t$ or $f'(x)dx = dt$ or $g(x)dx = dt$
==> $I = \int_{f(a)}^{f(b)} {tdt = \left| {\frac{{{t^2}}}{2}} \right|_{f(a)}^{f(b)}} = \frac{{{{[f(b)]}^2} - {{[f(a)]}^2}}}{2}$.

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