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

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

Answers (1)

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

Solution:

\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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