# int (2x^2 + 7x - 3)/(x - 2) dx Find the indefinite integral.

int (2x^2+7x-3)/(x-2)dx

To solve, divide the numerator by the denominator (see attached figure).

= int (2x + 11 + 19/(x-2)) dx

Express it as sum of three integrals.

= int 2xdx + int11dx + int 19/(x-2)dx

For the first integral, apply the formula int x^ndx = x^(n+1)/(n+1)+C .

For the second integral, apply the formula int adx = ax + C .

= (2x^2)/2 + 11x + C + int 19/(x-2)dx

=x^2+11x+C + int 19/(x-2)dx

For the third integral, use u-substitution method.

Let,

u = x - 2

Differentiate u.

du = dx

Then, plug-in them to the third integral.

=x^2+11x+C+19int 1/(x-2)dx

=x^2+11x+C+19int 1/udu

To take the integral of it, apply the formula int 1/xdx =ln|x| +C .

= x^2+11x + 19ln|u| + C

And substitute back u = x-2 .

=x^2+11x+19ln|x-2|+C

Therefore, int (2x^2+7x-3)/(x-2)dx = x^2+11x + 19ln|x-2| + C .

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)
Approved by eNotes Editorial Team