Problem 9

Question

Find the indefinite integral. $$ \int \frac{x^{3}-3 x^{2}+5}{x-3} d x $$

Step-by-Step Solution

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Answer

The integral of the given function is $x^3/3 + 3x^2/2 + 14x - 42 \ln|x - 3| + C$, where $C$ is the constant of integration.

1Step 1: Polynomial Division

Divide the numerator polynomial with the denominator in the integral, which gives us $x^2 + 3x + 14 - 42/(x - 3)$.

2Step 2: Solving the Individual Integrals

Now we find the integral of each individual term separately. The integral of $x^2$ is $x^3/3$, the integral of $3x$ is $3x^2/2$ the integral of $14$ is $14x$, and the integral of $-42/(x - 3)$ is $-42 \ln|x - 3|$.

3Step 3: Combining the Integrals

Combine all individual integrals with the respective coefficients to get the final result.

Problem 9

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