Integral Calculator — Step-by-Step (Substitution / By Parts / Partial Fractions)

Real-world use cases

• Verify a u-substitution problem before turning it in

  The textbook said ∫ 2x·cos(x²) dx = sin(x²) + C. You got there but cannot quite explain why you set u = x². Paste 2*x * cos(x^2) into the calculator. The step panel writes "let u = x², du = 2x dx, so the ∫ collapses to ∫ cos(u) du = sin(u), back-substitute u → sin(x²)". You see the mechanical reason: the 2x out front is literally what du looks like, so the substitution turns a hard integrand into a trivial one. Two minutes to internalize the trick that will save you on the next quiz problem.

• Practice integration by parts on x·exp(x)

  The classic by-parts exercise. Type x * exp(x). The step panel reads "LIATE: u = x (algebraic), dv = exp(x) dx, so du = dx, v = exp(x), then u·v − ∫v du = x·exp(x) − ∫exp(x) dx = x·exp(x) − exp(x)". You can copy each piece to your worksheet and tick it off. The verified badge below confirms d/dx of x·exp(x) − exp(x) really is x·exp(x), so your answer is consistent — not just the right shape.

• Solve a partial-fractions integral the page recognizes

  1/(x²−4) is the textbook intro to partial fractions. Type 1 / (x^2 - 4). The page detects the (x²−a²) form, writes out 1/((x−2)(x+2)) = (1/4)(1/(x−2) − 1/(x+2)), integrates each piece, and prints (1/4) · ln|(x−2)/(x+2)| + C. You see why a = 2 mattered, why the coefficient came out to 1/(2a), and how the two log terms combine into a single ratio inside the absolute value. The plot makes the asymptotes at x = ±2 obvious.

• Get a definite integral value without doing the arithmetic

  Once F(x) is computed, the definite-integral panel asks for a and b. Type 0 and pi, hit enter — the page evaluates F(pi) − F(0) numerically. For ∫₀^π sin(x) dx = 2 this lines up with the known answer; for trickier antiderivatives you can sanity-check against Wolfram or your CAS. No need to recopy F(x) into a separate calculator and risk a sign error in the transcription.

• See an honest "no elementary form" message and learn why

  Type exp(-x^2). The page returns an orange panel: "Could not integrate — some integrals truly have no elementary form (e.g. ∫ exp(−x²) dx). Try a different substitution by hand, or use a numerical method for definite integrals." This is not a calculator bug — it is Liouville's theorem. The Gaussian integrand has no antiderivative expressible in elementary functions; the function erf was invented to name its antiderivative. Seeing the failure framed correctly is how students learn the boundary of what symbolic integration can do.

Common pitfalls

• Not every integral has an elementary antiderivative. ∫ e^(−x²) dx, ∫ sin(x)/x dx, ∫ 1/ln(x) dx, ∫ e^x/x dx, ∫ √(1+x³) dx — Liouville's theorem (1835) proves these cannot be written with elementary functions. The calculator will say so honestly rather than fabricate an answer; do not assume a failure means you typed it wrong.

• Writing implicit multiplication. "2x" and "sin x" do not parse — use 2*x and sin(x). The error pinpoints the character position where the parser stopped so you do not have to hunt.

• Forgetting the + C. The page prints "+ C" next to the answer as a reminder. An indefinite integral is a family of functions differing by a constant; leaving off + C costs marks even when the antiderivative is exactly right. Definite integrals don't need it because the constant cancels in F(b) − F(a).

• Confusing the constant of integration with the constant inside the integrand. ∫ 5 dx = 5x + C (the 5 is a coefficient on x), not 5 + C. The calculator handles this correctly, but it is the most common error on the second day of class.

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