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Integrals — proven by differentiating back

Type an integrand. The solver finds the antiderivative with named steps, then does the one check that settles it: it differentiates its own answer and shows it matching your original function.

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  • ✓Every answer verified before you see it

In short

An integral calculator finds the antiderivative: a function whose derivative is your integrand. VerifySolve integrates term by term using the reverse power rule and linear substitution, showing each step. Then it verifies the result the strongest way possible — by differentiating the antiderivative and checking it reproduces exactly the function you started with.

The integration rules

Integration is differentiation read backwards. These are the reverse rules the engine uses — including the substitution pattern that handles anything with a linear inside.

Reverse power rule

∫xn dx=xn+1n+1+C(n≠−1)\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C \quad (n \ne -1)

The n = −1 exception

∫1x dx=ln⁡∣x∣+C\int \dfrac{1}{x} \, dx = \ln|x| + C

Trig

∫sin⁡x dx=−cos⁡x+C∫cos⁡x dx=sin⁡x+C\int \sin x \, dx = -\cos x + C \qquad \int \cos x \, dx = \sin x + C

Exponential

∫ex dx=ex+C\int e^x \, dx = e^x + C

Linear substitution

∫f(ax+b) dx=1aF(ax+b)+C\int f(ax + b) \, dx = \dfrac{1}{a} F(ax + b) + C

Worked examples — each differentiated back

The note under each answer is the verification the live solver performs: differentiate the result and compare with the integrand.

∫ 3x² dx

  1. 1.Reverse power rule: raise the power by 1, divide by the new power
  2. 2.3 · x³/3

x³ + C

Differentiate back: d/dx [x³] = 3x² ✓ — the original integrand.

∫ (x² + 2x) dx

  1. 1.Integrate term by term: x² → x³/3, 2x → x²

x³/3 + x² + C

Differentiate back: x² + 2x ✓.

∫ cos(2x) dx

  1. 1.Substitution with u = 2x (linear inside): the antiderivative of cos is sin
  2. 2.Scale by 1/2 to undo the chain rule’s factor of 2

sin(2x)/2 + C

Differentiate back: (cos(2x) · 2)/2 = cos(2x) ✓.

∫ 1/x dx

  1. 1.This is the n = −1 exception — the power rule cannot produce it

ln|x| + C

Differentiate back: d/dx [ln x] = 1/x ✓.

∫ e^(3x) dx

  1. 1.Exponential with a linear inside: e^(3x) keeps its shape
  2. 2.Scale by 1/3 for the inside derivative

e^(3x)/3 + C

Differentiate back: 3e^(3x)/3 = e^(3x) ✓.

Common mistakes — and how to dodge them

Forgetting the + C

An indefinite integral is a family of functions, not one function. Leaving off + C is technically wrong and costs marks. It is also a useful reminder of what integration cannot tell you without an extra condition.

Power-ruling 1/x

Applying the reverse power rule to x⁻¹ gives x⁰/0 — division by zero, the formula’s way of refusing. 1/x is the special case: its antiderivative is ln|x|. The exception exists precisely because the rule breaks.

Missing the 1/a in substitution

∫cos(2x) dx is sin(2x)/2, not sin(2x). Differentiate your answer mentally: the chain rule spits out a 2, so the antiderivative must pre-divide by 2. Our checker differentiates for real — that is how it never misses the factor.

Integral Calculator — questions

How is an integral answer verified?

By differentiating it. Integration and differentiation are inverse operations, so if the derivative of our antiderivative equals your original integrand, the integral is correct — provably. The calculator differentiates its own answer symbolically and compares it with your integrand at several real points, and shows you the check.

Which integrals are covered?

Indefinite integrals of polynomials (reverse power rule), 1/x, sin, cos, exp and square roots — including constant multiples and linear-inside forms like cos(2x), e^(3x) and (3x + 1)³ handled by substitution. Every step names the rule used.

What is not covered yet?

Definite integrals (with numeric bounds), integration by parts (∫ ln x dx, ∫ x·eˣ dx), trig powers and partial fractions. Type one and the calculator explains the gap honestly — it never dresses up a guess as an antiderivative.

Why does every answer end with + C?

Because infinitely many functions share the same derivative — they differ only by a constant. The + C stands for whichever constant your problem’s conditions would pin down. Dropping it is the most common integral mistake there is.

What is the 1/a factor in answers like sin(2x)/2?

Undoing the chain rule: differentiating sin(2x) produces an extra factor of 2, so the antiderivative must divide by 2 to compensate. Any linear inside (ax + b) contributes a 1/a scale — the steps show where it comes from.

Is the integral calculator free?

Yes — free, no sign-up, no app, steps and verification included.

Keep solving

Derivative Calculator →

The forward direction — rule-named steps, slope-verified.

Implicit Differentiation Calculator →

dy/dx for relations you cannot solve for y first.

Exponent Calculator →

Power rules and exponential equations, exactly solved.

Algebra Calculator →

Equations, systems and factoring with plug-back proof.

A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.