Integral Calculator
Calculus
Intro: Enter an integral like $\int \! f(x)\,dx$ or a definite form $\int_a^b f(x)\,dx$. We show each rule used and verify by differentiating the result.
Worked example
- Compute $\displaystyle \int \dfrac{2x+3}{x^2+3x+2}\,dx$
- Factor denominator: $x^2+3x+2=(x+1)(x+2)$.
- Partial fractions: write $\dfrac{2x+3}{(x+1)(x+2)}=\dfrac{A}{x+1}+\dfrac{B}{x+2}$.
- Solve for $A,B$: $2x+3=A(x+2)+B(x+1)=(A+B)x+(2A+B)$. Match coefficients: $A+B=2$ and $2A+B=3$.
- Subtract equations: $(2A+B)-(A+B)=3-2 \Rightarrow A=1$, then $B=1$.
- Integral splits: $\int \left(\dfrac{1}{x+1}+\dfrac{1}{x+2}\right)dx=\ln|x+1|+\ln|x+2|+C$.
- Combine logs if desired: $\ln|(x+1)(x+2)|+C=\ln|x^2+3x+2|+C$.
- Check by differentiation: $\dfrac{d}{dx}[\ln|x+1|+\ln|x+2|]=\dfrac{1}{x+1}+\dfrac{1}{x+2}=\dfrac{2x+3}{x^2+3x+2}$ ✔️.
FAQs
Do you support partial fractions?
Yes—linear and many irreducible quadratic cases with full steps.
How do you verify?
We differentiate the antiderivative to recover the integrand.
Why choose Mathix?
- 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.