Partial Fractions Decomposition
Calculus, Algebra
Intro: Perfect for integrals that need decomposition first.
Worked example
- $\displaystyle \frac{2x+5}{(x-1)(x+2)}$
- Assume $\dfrac{2x+5}{(x-1)(x+2)}=\dfrac{A}{x-1}+\dfrac{B}{x+2}$.
- Combine: $2x+5=A(x+2)+B(x-1)=(A+B)x+(2A-B)$.
- Match coefficients: $A+B=2$ and $2A-B=5$.
- Solve: add equations: $3A=7\Rightarrow A=\tfrac{7}{3}$. Then $B=2-A=2-\tfrac{7}{3}=-\tfrac{1}{3}$.
- Decomposition: $\boxed{\tfrac{7/3}{x-1}-\tfrac{1/3}{x+2}}$.
FAQs
Handles repeated factors?
Yes — it sets up $\dfrac{A_1}{(x-a)}+\dfrac{A_2}{(x-a)^2}+\cdots$ automatically.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.