What it does
The hard part of integration is not the algebra but the first decision: which method? This coach looks at the integrand the way a teacher would and tells you which method fits and why, then works it through step by step.
- Power rule for polynomials and powers of x (with ln|x| for 1/x).
- Standard forms, including a linear inside such as sin(3x + 1) or 1/(2x + 3), where you divide by the inside’s slope.
- Substitution when an inside function sits there and the rest is a constant times its derivative, as in 2x cos(x²) or ln x / x.
- Integration by parts for a product of two different kinds, such as x eˣ or x ln x, with u chosen by ILATE (Inverse trig, Log, Algebraic, Trig, Exponential). A lone ln x or tan⁻¹ x is done by parts with dv = dx.
- Partial fractions for a fraction of polynomials: divide first if the top is too big, factor the bottom, match coefficients.
- Parts twice, then solve for eˣ sin x and eˣ cos x, where the integral comes back.
Every answer is checked by differentiating it back. The graph then draws f and its antiderivative F + C, with the tangent to F at a point you choose: its slope is always the height of f.
It follows integrals in Class 12 maths (methods of integration). It is the reverse of differentiation, so derivative steps is the best warm-up; afterwards, definite integral properties turns these antiderivatives into definite integrals, and riemann sums shows what the area really is.
How to use it
- Type the integrand between ∫ and dx, or tap an example. Examples are grouped by the method they need, so you can practise one method at a time.
- Read the method and the why. Try to guess the method first, then compare.
- Follow the steps. A sum is split into terms; each term’s own working folds out underneath.
- The answer ends in + C, and the green line confirms it differentiates back.
- In the graph, slide x to move the tangent, and C to slide the whole antiderivative up and down: every C gives a correct answer.
Worked example
∫ x eˣ dx. There are two different kinds of factor: x is algebraic, eˣ is exponential. In ILATE, A comes before E, so u = x and dv = eˣ dx. Then du = dx and v = eˣ, and
∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C.
At x = 1.5 the graph shows F(1.5) = 1.5e^1.5 − e^1.5 = 0.5 × 4.4817 = 2.2408, and the slope of F there is 6.7225, which is exactly f(1.5) = 1.5 × 4.4817.
Now tap (3x + 5)/(x² + 3x + 2). The bottom factors as (x + 1)(x + 2); matching coefficients gives 2/(x + 1) + 1/(x + 2), so the answer is 2 ln|x + 1| + ln|x + 2| + C.
FAQ
Why did it choose substitution and not parts? If an inside function and its derivative both appear, substitution is quicker. ∫ x e^(x²) dx by parts goes round in circles; with u = x² it is one line.
What if it says no school method fits? Some integrals, such as e^(x²) or sin(x)/x, have no answer in ordinary functions at all. Others need tricks beyond this coach; the symbolic algebra tool tries a little more.
Is this page part of NCERT? No. fibub is independent. It follows the topics of the Class 12 maths syllabus and uses none of the textbook’s text or exercises.
Where the numbers come from
- Standard school integration methods: the power rule, standard integrals, substitution, integration by parts with the ILATE order, partial fractions with linear and quadratic factors. Written for fibub; no textbook text, figures or exercises are used.
- Checked against sympy 1.14 (integrate): definite integrals from 1.2 to 1.9 of 27 school integrands, plus a check that every answer differentiates back to its integrand.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.