Problems on Techniques of Integration

Since the degree of the numerator is bigger than the degree of the denominator, we will perform the long division to get

$$\frac{x^n}{ax^2 + b x + c}= \frac{1}{a} x^{n-2} - \frac{1}{a}\;\;\frac{bx^{n-1} + cx^{n-2}}{ax^2 + b x + c}$$

So

$$\int \frac{x^n}{ax^2 + b x + c}dx = \frac{1}{a(n-1)}x^{n-1} - \frac{1}{a}\; \int \frac{bx^{n-1} + cx^{n-2}}{ax^2 + b x + c}dx$$

or

$$\int \frac{x^n}{ax^2 + b x + c}dx = \frac{1}{a(n-1)}x^{n-1} -... ...b x + c}dx - \frac{c}{a}\;\int \frac{x^{n-2}}{ax^2 + b x + c}dx$$

Detailed Answer.

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