Free Calculus Solver
Type any derivative, integral, limit or differential equation and see the full step-by-step working — chain rule, integration by parts, L'Hôpital and beyond.
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Integration by parts — a worked example
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMO- 1
Recognise the form and pick the technique
The integrand is a product of an algebraic function (x) and an exponential (eˣ). Integration by parts is the go-to: ∫u dv = uv − ∫v du.
- 2
Choose u and dv (LIATE rule)
Logs/Algebraic beat Trig/Exp when picking u — so u = x here.
- 3
Apply the integration-by-parts formula
- 4
Evaluate the remaining integral
- 5
Combine and add the constant
Quick check: differentiate (x − 1)eˣ → eˣ + (x − 1)eˣ = x eˣ. ✓
Final Answer
How to solve a calculus problem — derivatives, integrals and limits
Calculus studies how things change (differentiation) and how changes accumulate (integration). Differentiation gives the slope of a curve or the instantaneous rate of change; the key rules — power rule, product rule, quotient rule and chain rule — cover almost everything in AP Calculus AB and Class 11. Integration reverses differentiation; the key techniques are substitution, integration by parts, partial fractions and trigonometric substitutions, all standard in A-Level, AP Calc BC and JEE.
Limits sit underneath both, and L'Hôpital's rule, squeeze theorem and series expansions are the usual toolkit when you hit 0/0 or ∞/∞. Calculus also extends into differential equations (first-order separable, linear, homogeneous) and classic applications — optimisation, related rates, area between curves, volumes of revolution. The solver above shows the full working for each, including which rule or technique it picked and why.
Calculus problems to practise (AP Calc, A-Level, Class 11–12)
Tap any problem to solve it with full step-by-step working.
- Solve with AI →1.Power ruleClass 11 / AP ABEasy
- Solve with AI →2.Chain ruleClass 12 / AP ABMedium
- Solve with AI →3.Standard integralA-Level / AP BCMedium
- Solve with AI →4.Limit (L'Hôpital)Class 12 / JEEMedium
- Solve with AI →
5. Find the area enclosed between y = x² and y = 2x for 0 ≤ x ≤ 2.
Area between curvesAP BC / A-LevelHard
Frequently asked questions
How do I pick the right differentiation rule?+
Use the power rule for polynomials, chain rule when a function is nested inside another (sin(3x²), (x² + 1)⁵), product rule for explicit products (x² sin x), and quotient rule for fractions. SolveGini picks the rule for you and shows why.
What's the best technique for integration by parts?+
Use the LIATE mnemonic for choosing u: Logarithmic, Inverse trig, Algebraic, Trig, Exponential — whichever appears first in that list becomes u. For ∫x eˣ dx, x is Algebraic and beats Exponential, so u = x.
When does L'Hôpital's rule apply?+
Only for indeterminate forms 0/0 or ∞/∞. Differentiate numerator and denominator separately and re-evaluate the limit. If you hit another 0/0, apply L'Hôpital again — but check for algebraic simplification first, it's often faster.
Does the calculus solver handle JEE-level problems?+
Yes. Definite integrals with trigonometric substitutions, parametric differentiation, higher-order derivatives, and the standard JEE Main + Advanced chapters (application of derivatives, integrals, differential equations) are all in scope with step-by-step working.
Is the calculus solver free to use?+
SolveGini has a free plan. Guests can try 1 solve per day; a free account unlocks 5 daily solves plus quizzes, flashcards and the study planner — step-by-step working is never paywalled.
Can I use this for AP Calculus AB and BC preparation?+
Yes. AP AB (limits, derivatives, integrals, FTC) and AP BC (series, parametric, polar, advanced integration) are both covered. Practice in the problem format the College Board uses, with full working shown for each step.
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Related topics to master next
Calculus builds on algebra and trigonometry — keep those sharp.