Integrals — Part 1: Antiderivatives, substitution and special forms: visual lesson, CBSE Class 12 Mathematics

Lesson

Integrals, Part 1

Run differentiation backwards: from a slope, recover the curve.

In this lesson you will

Every curve $y = x^2 + C$ has slope $2x$: one derivative, a whole family of antiderivatives.

Concept

Concept

If $\dfrac{d}{dx}F(x) = f(x)$ , then $F$ is an antiderivative (primitive) of $f$ , and we write $\displaystyle\int f(x)\,dx = F(x) + C$ .

\frac{d}{dx}\left(x^3\right) = 3x^2 \;\Longrightarrow\; \int 3x^2\,dx = x^3 + C