What Are Integrals and How Do You Solve Them?
An integral measures accumulation: it adds up infinitely many infinitesimal pieces to produce a total. There are two main types. An indefinite integral gives a family of antiderivative functions and always includes a constant of integration, written as ∫f(x)dx = F(x) + C. A definite integral gives a single number by evaluating that antiderivative between two limits, written as ∫ₐᵇ f(x)dx = F(b) − F(a). You solve indefinite integrals by applying integration rules and methods; you solve definite integrals the same way, then subtract the antiderivative values at the endpoints. Tools like MathDF can compute both and show the steps, which is useful for checking your own work.
Indefinite vs. definite integrals
| Indefinite integral | Definite integral | |
|---|---|---|
| Notation | ∫f(x)dx | ∫ₐᵇ f(x)dx |
| Result | A function family, F(x) + C | A single number |
| Role of C | Required — infinitely many antiderivatives differ by a constant | Cancels out during evaluation |
| Main use | Finding antiderivatives, setting up further work | Computing areas, totals, accumulated change |
The constant C matters only for indefinite integrals. When you evaluate a definite integral, F(b) − F(a) removes it, so you never write + C in the final answer.
Basic integration rules
These cover most straightforward problems:
- Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, valid for n ≠ −1. For n = −1, ∫(1/x)dx = ln|x| + C.
- Constant multiple: ∫k·f(x)dx = k·∫f(x)dx.
- Sum rule: ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx.
- Exponential: ∫eˣ dx = eˣ + C; ∫aˣ dx = aˣ/ln(a) + C.
- Trigonometric: ∫sin(x)dx = −cos(x) + C; ∫cos(x)dx = sin(x) + C; ∫sec²(x)dx = tan(x) + C.
For example, ∫(3x² + 2x)dx = x³ + x² + C, applying the power rule and sum rule term by term.
Choosing a method: substitution vs. integration by parts
When a basic rule does not fit, two methods handle most cases:
Substitution works when the integrand contains a function and (a multiple of) its derivative. Set u equal to the inner function, rewrite dx in terms of du, integrate, then substitute back. Example: ∫2x·cos(x²)dx — let u = x², so du = 2x dx, giving ∫cos(u)du = sin(u) + C = sin(x²) + C.
Integration by parts works for products of different function types, especially polynomial × exponential, polynomial × trig, or polynomial × logarithm. The formula is ∫u dv = uv − ∫v du. Choose u to be the part that simplifies when differentiated. Example: ∫x·eˣ dx — let u = x, dv = eˣ dx, so du = dx and v = eˣ, giving x·eˣ − ∫eˣ dx = x·eˣ − eˣ + C.
A quick rule of thumb: if you can see a function and its derivative inside the integrand, try substitution first. If you have a product where neither factor is the other's derivative, try parts.
Solving integrals with a step-by-step tool
MathDF accepts a problem typed directly, built with its math keyboard, photographed, or drawn by hand. According to the site, every change to the input is analyzed in the browser, and if it contains a math expression the built-in calculator solves it instantly under the field before anything is sent to the AI. When a problem can be solved step by step, a link button lights up and opens the matching calculator with your expression already loaded.
To solve an integral:
- Enter the expression. Type it, use the math keyboard (which includes integral signs, fractions, roots, powers, and limits), or photograph/draw the problem. Recognition converts a photo or sketch into an expression in the input, where you can fix a symbol before solving.
- Read the instant result. Roots, simplified forms, derivatives, integrals, limits, and exact and decimal values appear under the input. You can tap a result to insert it back into the field.
- Open the step-by-step solution. If the problem is solvable step by step, use the link button to load the matching calculator with your expression.
- Send to the AI chat if needed. Text and images go to the assistant for explanations, word problems, proofs, or checking your own solution. There are two modes — a math-oriented tutor and a general assistant — and you can switch at any time.
The site also offers ready-made examples: tapping one inserts its expression into the input.
Common mistakes to watch for
- Forgetting + C on an indefinite integral. The answer is incomplete without it.
- Dropping the absolute value in ∫(1/x)dx = ln|x| + C.
- Mishandling limits in substitution. When you change variables in a definite integral, either change the limits to match u or substitute back to x before evaluating.
- Sign errors in integration by parts. The formula subtracts the second integral: uv − ∫v du, not uv + ∫v du.
- Assuming every integral has a closed form. Some integrals cannot be expressed with elementary functions, and no method will produce one.
FAQ
Do I always need the constant C? Only for indefinite integrals. In a definite integral it cancels during evaluation.
When should I use substitution instead of parts? Use substitution when the integrand contains a function and its derivative. Use parts for products of different function types where neither factor is the other's derivative.
Can I check my answer? Yes. Differentiate your antiderivative — if you get back the original integrand, the indefinite integral is correct. For definite integrals, you can also verify numerically.
Can I solve integrals from a photo? MathDF supports photo and handwriting recognition, converting the image into an expression you can edit before solving.