To evaluate the integral

$\int \frac{x + 3}{2x^2 - 9x + 10} \, dx,$

we can use partial fraction decomposition to simplify the integrand if possible. Here's the step-by-step process:

1. Factorize the Denominator

The denominator $2x^2 - 9x + 10$ can be factored: $2x^2 - 9x + 10 = (2x - 5)(x - 2).$

So, the integrand becomes: $\frac{x + 3}{2x^2 - 9x + 10} = \frac{x + 3}{(2x - 5)(x - 2)}.$

2. Decompose into Partial Fractions

We express: $\frac{x + 3}{(2x - 5)(x - 2)} = \frac{A}{2x - 5} + \frac{B}{x - 2}.$

Multiply through by the denominator $(2x - 5)(x - 2)$: $x + 3 = A(x - 2) + B(2x - 5).$

Expand and collect terms: $x + 3 = A x - 2A + 2B x - 5B.$

Combine like terms: $x + 3 = (A + 2B)x - (2A + 5B).$

Equate coefficients of $x$ and the constant terms:

• Coefficient of $x$: $A + 2B = 1$,
• Constant term: $-2A - 5B = 3$.

3. Solve for $A$ and $B$

From $A + 2B = 1$, solve for $A$: $A = 1 - 2B.$

Substitute $A = 1 - 2B$ into $-2A - 5B = 3$: $-2(1 - 2B) - 5B = 3,$ $-2 + 4B - 5B = 3,$ $-2 - B = 3,$ $B = -5.$

Substitute $B = -5$ into $A = 1 - 2B$: $A = 1 - 2(-5) = 1 + 10 = 11.$

So: $A = 11, \quad B = -5.$

4. Rewrite the Integral

The integrand becomes: $\frac{x + 3}{(2x - 5)(x - 2)} = \frac{11}{2x - 5} - \frac{5}{x - 2}.$

The integral is: $\int \frac{x + 3}{2x^2 - 9x + 10} \, dx = \int \frac{11}{2x - 5} \, dx - \int \frac{5}{x - 2} \, dx.$

5. Evaluate the Integrals

Each term can be integrated: $\int \frac{11}{2x - 5} \, dx = \frac{11}{2} \ln|2x - 5|,$ $\int \frac{5}{x - 2} \, dx = 5 \ln|x - 2|.$

So the final result is: $\int \frac{x + 3}{2x^2 - 9x + 10} \, dx = \frac{11}{2} \ln|2x - 5| - 5 \ln|x - 2| + C,$ where $C$ is the constant of integration.

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Would you like a deeper explanation of any step or help applying this to a similar problem? 😊

Related Questions

1. How do you factorize quadratic expressions like $2x^2 - 9x + 10$?
2. What is partial fraction decomposition, and why is it useful?
3. How do you solve systems of linear equations like the one for $A$ and $B$?
4. What are common mistakes to avoid when finding partial fractions?
5. Can this method be extended to higher-degree polynomials?

Tip

When performing partial fraction decomposition, always double-check your factorization and coefficients before proceeding to integration!