Question

Evaluate ∫limitsx²/(x²+4)(x²+9) .

Answer 1

The integral of x² over (x² + 4)(x² + 9) is arctan(x/2) - (1/4) * arctan(3x/4) + C, where C is the constant of integration.

Explanation:

To solve the integral ∫x² / (x² + 4)(x² + 9) dx, we first decompose the denominator into partial fractions. Let's express the integrand as A/(x² + 4) + B/(x² + 9). Upon solving for A and B, we get A = 1/5 and B = -1/5.

Now, we integrate each term separately. The integral of 1/(x² + a²) is arctan(x/a) * (1/a) + C, where C is the constant of integration. Applying this formula to both terms gives us (1/5) * arctan(x/2) - (1/5) * arctan(3x/4).

Combining these results, the final answer to the integral becomes arctan(x/2) - (1/4) * arctan(3x/4) + C, where C is the constant of integration. This solution represents the integral of x² over (x² + 4)(x² + 9). The arctan function arises due to the integration of inverse trigonometric functions resulting from the partial fraction decomposition, and the constants are derived from the integration of each term.

Answer 2

Main Answer:

The integral of
`x²/(x²+4)(x²+9)` can be evaluated using partial fraction decomposition.

Explanation:

To evaluate the given integral, we employ partial fraction decomposition. First, express the rational function as a sum of simpler fractions, where the denominators are irreducible. In this case, factorize the denominator into
`(x²+4)(x²+9)`. The partial fraction decomposition involves finding constants for each irreducible factor. In this scenario, the irreducible factors are
`(x²+4) and (x²+9)`. Set up an equation with unknown constants for each factor and solve for their values.

Once the constants are determined, rewrite the original expression as a sum of the partial fractions. This allows us to integrate each fraction separately, making the overall integration more manageable. Integrate each term individually and combine the results. The integral should now be expressed in terms of natural logarithms and arctangent functions.

The process of partial fraction decomposition simplifies the integration of complex rational functions, breaking them down into simpler components. This method is especially useful when dealing with higher-degree polynomials in the denominator. In conclusion, by decomposing the given function into partial fractions and integrating each term separately, the result can be expressed in terms of logarithmic and arctangent functions.