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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 56

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Answer: \frac{e^{x}}{x}+c

Hints: You must know about the rules of exponential function of integration.

Given: \int \frac{\left ( x-1 \right )}{x^{2}}e^{x}dx=f\left ( x \right )e^{x}+c


\begin{aligned} &\int \frac{(x-1)}{x^{2}} e^{x} d x \\ &\int e^{x} f(x)+f^{1}(x) d x \\ &e^{x} f(x)+c \quad\quad\quad\quad\quad\quad\left\{\int e^{x}\left(f(x)+f^{1}(x)\right) d x=e^{x} f(x)+c\right\} \end{aligned}

\begin{aligned} &\frac{x-1}{x^{2}}=\frac{x}{x^{2}}-\frac{1}{x^{2}} \\ &=\frac{1}{x}-\frac{1}{x^{2}} \\ &f(x)=\frac{1}{x}, f^{1}(x)=\frac{-1}{x^{2}} \end{aligned}
\begin{aligned} &f(x)+f^{1}(x)=\frac{1}{x}-\frac{1}{x^{2}} \\ &\therefore I=\int e^{x}\left(f(x)+f^{1}(x)\right) d x \quad \quad \quad \text { if } f(x)=\frac{1}{x} \\ &I=e^{x} f(x)+c \\ &\therefore f(x)=\frac{1}{x} \\ &I=e^{x} / x+c \end{aligned}

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