What an Integral Is and Why +C Appears
An indefinite integral reverses differentiation. ∫f(x) dx is the family of functions whose derivative is f(x). Because constants vanish when you differentiate, x² + 1 and x² − 40 share the derivative 2x, and the +C records that whole family.
That constant is not decoration. If velocity is v = 3t², then position is t³ + C, and C is where the object started. Drop it and you lose the starting point.
Choosing the Right Method
| If the integrand looks like | Try | Example |
|---|---|---|
| A polynomial or power | Power rule | ∫x⁴ dx = x⁵/5 + C |
| A function times its inner derivative | u-substitution | ∫2x cos(x²) dx = sin(x²) + C |
| A product of unlike types (x·eˣ, x·sin x, ln x) | Integration by parts | ∫x eˣ dx = (x − 1)eˣ + C |
| A ratio of polynomials | Partial fractions | ∫1/(x² − 1) dx = ½ ln|(x−1)/(x+1)| + C |
| sin² or cos² | Half-angle identity | ∫sin² x dx = x/2 − sin 2x/4 + C |
For parts, the LIATE order (Logarithm, Inverse trig, Algebraic, Trig, Exponential) tells you what to call u. It works about nine times in ten and the calculator uses it.
Worked Examples
∫x²eˣ dx
- Let u = x², dv = eˣ dx. Then du = 2x dx, v = eˣ, giving x²eˣ − ∫2x eˣ dx.
- Repeat on ∫2x eˣ dx with u = 2x: it equals 2x eˣ − 2eˣ.
- Combine: x²eˣ − 2x eˣ + 2eˣ.
Answer: eˣ(x² − 2x + 2) + C
∫1/(x² − 1) dx
- Factor: x² − 1 = (x − 1)(x + 1).
- Split: 1/(x² − 1) = ½[1/(x − 1) − 1/(x + 1)].
- Integrate each piece into a logarithm.
Answer: ½ ln|x − 1| − ½ ln|x + 1| + C
∫sin² x dx
- Use sin² x = (1 − cos 2x) ÷ 2.
- Integrate: x/2 − sin(2x)/4.
Answer: x/2 − sin(2x)/4 + C
When There Is No Closed Form
Some integrands look harmless but have no antiderivative made of standard functions. Famous ones are e−x², sin x ÷ x, 1/ln x and xx. Liouville proved in the 1830s that this is a true limit of mathematics, not of any calculator.
You can still get a number for a definite integral of these. The definite integral mode computes ∫ e−x² over the whole real line as 1.77245, which is √π, even though no formula for the antiderivative exists.
What We Think Is Overrated, Underrated and Wrong
Integrals People Search For
| Search | Answer | Method |
|---|---|---|
| ∫ 1/x dx | ln|x| + C | log rule |
| ∫ ln x dx | x ln x − x + C | by parts, u = ln x |
| ∫ sin²x dx | x/2 − sin(2x)/4 + C | half-angle identity |
| ∫ sec x dx | ln|sec x + tan x| + C | standard form |
| ∫ x eˣ dx | (x − 1)eˣ + C | by parts |
| ∫ 1/(1 + x²) dx | arctan x + C | standard form |
| ∫ e^(−x²) dx | (√π/2) erf(x) + C | no elementary antiderivative |
Why Three Different Answers Can All Be Right
For ∫ sin x cos x dx you can get sin²x ÷ 2, or −cos²x ÷ 2, or −cos(2x) ÷ 4. They look different but differ only by constants (for example sin²x ÷ 2 − (−cos²x ÷ 2) = 1/2), so all are correct once +C is attached. The reliable test is to differentiate your answer and compare it with the integrand.
Our Take: What Is Overrated, Underrated and Plain Wrong
Overrated: memorising a table of 30 integrals. Ten standard forms plus substitution and parts cover most courses.
Underrated: differentiating your result. It takes seconds and is the one check that never lies.
Plain wrong: dropping +C or writing ln x instead of ln|x|. Both lose valid answers.
Integral Mistakes That Cost Marks
- Forgetting +C on indefinite integrals.
- Writing ∫1/x dx = ln x without absolute value. It is ln|x| + C.
- Dropping the 1/a when integrating a linear inside: ∫cos(5x) dx = sin(5x)/5.
- Applying the product rule backwards: ∫fg dx is not ∫f dx × ∫g dx.
- Substituting u but forgetting to change dx.
Why Students Get Stuck on Integrals, and What a Good Calculator Should Do
Searching for help with integrals usually starts late at night with one problem that will not work out. There is no single rule, so choosing between substitution, parts and tables is the hard part. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.
In short: students need to recognise the pattern and finish with the + C, not just see a final number, and a calculator earns trust only when its steps can be checked.
Understand the pain points
What students get stuck on
Most mistakes here are not about arithmetic. They come from one rule applied the wrong way round, which is why seeing every step matters.
- Forgetting the constant of integrationIntegrate x^2 * e^x↗
- Choosing the wrong part for integration by partsIntegrate 2x * cos(x^2)↗
- Not changing back to x after a substitutionIntegrate 1/(x^2 + 1)↗
- Not checking by differentiating the answer
Understand the frustrations and desires
What they feel, and what they want
Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.
- Wants the method named before it is applied
- Wants the answer differentiated back as a check
- Wants simple cases and hard cases in one place
Build the page around their intent
Each intent has a place on this page
The page follows intent, not features, so each need has a clear starting point.
- I need an answer now: type the problem into the calculator at the top of this page.Integrate x^2 * e^x↗
- I need to understand: the sections below explain the idea in plain words and list the common mistakes.
- I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
- I need practice: open the solver, tap + and choose Create Practice Test.
- I need it explained aloud: in the solver, tap + and choose Create Video.