Question

integration x/a+x dx​

Answer 1

$\underline{\textsf{Solution :}}$

$\mathsf{Now,\:\int \frac{x\:dx}{a+x}}$

$\mathsf{=\int \frac{\{(a+x)-a\}dx}{a+x}}$

$\mathsf{=\int dx-a\int \frac{dx}{a+x}}$

$\mathsf{=x-a\:log|a+x|+C}$

$\textsf{where C is integral constant}$

$\to \boxed{\mathsf{\int \frac{x\:dx}{a+x}=x-a\:log|a+x|+C}}$

$\underline{\textsf{Integration formulas :}}$

$\mathsf{1.\:\int x^{n}dx=\frac{x^{n+1}}{n+1}+C}$

$\textsf{where C is integral constant}$

$\mathsf{2.\:\int sinmx\:dx=-\frac{cosmx}{m}+C}$

$\textsf{where C is integral constant}$

$\mathsf{3.\:\int cosmx\:dx=\frac{sinmx}{m}+C}$

$\textsf{where C is integral constant}$

$\mathsf{4.\:\int e^{mx}dx=\frac{e^{mx}}{m}+C}$

$\textsf{where C is integral constant}$

$\mathsf{5.\:\int \frac{dx}{x+a}=log|x+a|+C}$

$\textsf{where C is integral constant}$

Answer 2

Answer:

$→\int \frac{x}{(x + a)} dx \\ = \int \frac{(x + a) - a}{(x + a)} dx \\ = \int \frac{(x + a)}{(x + a)} dx - \int \frac{a}{(x + a)} dx \\ = \int dx - a \int \frac{dx}{(x + a)} \\ = x - a \: log|x + a| +C\\ = x - { log( |x + a| ) }^{a} + C$

• x-log(|x+a|)^a+C is the right answer.