Integrate (x2-5x+6) / (x−3) dx

Question

ScanSolve AI Tutor · Accepted Answer

The problem asks to evaluate the indefinite integral:

\int \left( x^2 - \frac{5}{x} + 3 \sec^2 x + 5 \sin 3x - \frac{6}{x} \right) dx

Combine like terms in the integrand.
The terms $-\frac{5}{x}$ and $-\frac{6}{x}$ can be combined.
$-\frac{5}{x} - \frac{6}{x} = -\frac{11}{x}$
So the integral becomes:
$\int \left( x^2 - \frac{11}{x} + 3 \sec^2 x + 5 \sin 3x \right) dx$
Integrate each term separately using the standard integration rules.
Recall the rules:
- $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$ )
- $\int \frac{1}{x} dx = \ln|x| + C$
- $\int \sec^2 x dx = \tan x + C$
- $\int \sin(ax) dx = -\frac{1}{a} \cos(ax) + C$
Integrate $x^2$ .
$\int x^2 dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3}$
Integrate $-\frac{11}{x}$ .
$\int -\frac{11}{x} dx = -11 \int \frac{1}{x} dx = -11 \ln|x|$
Integrate $3 \sec^2 x$ .
$\int 3 \sec^2 x dx = 3 \int \sec^2 x dx = 3 \tan x$
Integrate $5 \sin 3x$ .
$\int 5 \sin 3x dx = 5 \left( -\frac{1}{3} \cos 3x \right) = -\frac{5}{3} \cos 3x$
Combine all the integrated terms and add the constant of integration, $C$ .
$\frac{x^3}{3} - 11 \ln|x| + 3 \tan x - \frac{5}{3} \cos 3x + C$
The final answer is:
$\boxed{\frac{x^3}{3} - 11 \ln|x| + 3 \tan x - \frac{5}{3} \cos 3x + C}$

Integrate (x2-5x+6) / (x-3) dx