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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Evaluate: $\int \frac{x-4}{(x-2)^3} \cdot e^x d x$
  • A
    $\frac{e^x}{(x-2)^3}+C$
  • B
    $\frac{-e^x}{(x-2)^3}+C$
  • C
    $\frac{e^x}{(x-2)^2}+C$
  • D
    $\frac{-e^x}{(x-2)^2}+C$
✓

Answer

$\begin{array}{l}\text { (c) }: \int \frac{x-4}{(x-2)^3} \cdot e^x d x=\int\left[\frac{x-2}{(x-2)^3}-\frac{2}{(x-2)^3}\right] e^x d x \\ =\int\left[\frac{1}{(x-2)^2}-\frac{2}{(x-2)^3}\right] e^x d x=\frac{e^x}{(x-2)^2}+C \\ {\left[\because \int\left[f(x)+f^{\prime}(x)\right] e^x d x=e^x f(x)+C\right]}\end{array}$

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