The method of Integration by Partial Fractions represents an essential calculus strategy designed to resolve intricate rational expressions. This approach is characterized by the breakdown of a complex fraction into more manageable fractions, which are then individually integrated. This technique proves particularly beneficial when faced with polynomial expressions found within denominators.
| Topic | Problem | Solution |
|---|---|---|
| None | Evaluate the integral $\int_{1}^{\sqrt{2}} \frac{… | Given the integral \(\int_{1}^{\sqrt{2}} \frac{s^{2}+\sqrt{s}}{s^{2}} d s\) |
| None | Find the inverse Laplace transform of the followi… | The given function is a product of two functions, \(\frac{1}{s}\) and \(\frac{1}{s^{2} + 16}\). |
| None | Find $\int \frac{x-1}{x^{2}} d x$ | Given the integral problem \(\int \frac{x-1}{x^{2}} dx\) |
| None | $\int \frac{5 x^{4}+7 x^{2}+x+2}{x\left(x^{2}+1\r… | \(\int \frac{5 x^{4}+7 x^{2}+x+2}{x\left(x^{2}+1\right)^{2}} d x = \int \frac{A}{x} + \frac{Bx + C}… |
| None | $\int \frac{10}{(x-1)\left(x^{2}+9\right)} d x$ | Rewrite the given function using partial fraction decomposition: $$\frac{10}{(x-1)(x^2+9)} = \frac{… |
| None | $\int \frac{2 x+1}{x^{2}-x-2} d x$ | First, let's find the partial fraction decomposition of the given function: $\frac{2x+1}{x^2-x-2} =… |
| None | $\int \frac{x^{3}+x}{x-1} d x$ | Perform polynomial long division to simplify the integrand: \(\frac{x^3 + x}{x - 1} = x^2 + x + 2 +… |
| None | $\int \frac{1}{x^{2}-1} d x$ | Perform partial fraction decomposition: \(\frac{1}{x^2 - 1} = \frac{1/2}{x - 1} - \frac{1/2}{x + 1}… |