Question

How do you evaluate this integral using substitution ?
∫x/x+3dx

Answer 1

\[u=x+3\implies x=?\]

Answer 2

it u=x+3 what does x equal ???

Answer 3

x= u-3 ?

Answer 4

dont go for substitution...do it like this......... (x+3)-3/(x+3)= (x+3) /(x+3) - 3/(x+3)


which inturn equals to 1- 3/(x+3)......and integration of 1 will be x.....and integration of 3/x+3 will be 3 ln(x+3)

Answer 5

By Substitution:
\[ t = x+3\]
\[ x = t - 3 \]
\[ dt = dx \]

Substitute them in the original equation:
\[ \int \frac{x}{x+3} dx \]
\[ = \int \frac{t-3}{t} dt \]
\[ = \int (1 - \frac{3}{t})dt \]
\[ = \int dt - \int \frac{3}{t} dt \]
\[ = t - 3ln(t) +c_1 \]
\[ = x + 3 - 3ln(x+3) + c_1 \]
\[ = x - 3ln(x+3) + c \]

Simple Solving:
\[ \int \frac{x}{x+3} dx \]
\[ = \int \frac{x+3-3}{x+3} dx \]
\[ = \int (1 - \frac{3}{x+3}) dx\]
\[ = \int dx - \int \frac{3}{x+3} dx \]
\[ = x - 3 ln(x+3) + c \]