The indefinite integral

Indefinite integral notation, the standard integrals of x^n and (ax+b)^n, and finding a function from its derivative and a known point.

Notation

An integral with no limits is an indefinite integral. It stands for every primitive, so it always includes a constant: $$\int f(x)\,dx = F(x) + C, \quad \text{where } F'(x) = f(x).$$

Standard integrals

Powers of $x$

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \ne -1$$

Powers of a linear function

$$\int (ax + b)^n\,dx = \frac{(ax + b)^{n+1}}{a(n + 1)} + C, \quad n \ne -1,\ a \ne 0$$

Example 1●

Find $\displaystyle\int (3x^2 - 4x + 5)\,dx$.

$x^3 - 2x^2 + 5x + C$

Example 2●●

Find (a) $\displaystyle\int (2x + 3)^4\,dx$ and (b) $\displaystyle\int \sqrt{4x + 1}\,dx$.

$\dfrac{(2x + 3)^5}{2 \times 5} + C = \dfrac{(2x + 3)^5}{10} + C$

a = 2, n = 4

$\dfrac{(4x + 1)^{\frac32}}{4 \times \frac32} + C = \dfrac{(4x + 1)^{\frac32}}{6} + C$

a = 4, n = 1/2

Example 3●●

Evaluate $\displaystyle\int_0^1 (2x + 1)^3\,dx$.

$\Big[\dfrac{(2x + 1)^4}{8}\Big]_0^1 = \dfrac{81 - 1}{8} = 10$

Using a known point

Example 4●●

$f'(x) = 6x^2 - 2$ and $f(1) = 5$. Find $f(x)$.

$f(x) = 2x^3 - 2x + C$.

$f(1) = 2 - 2 + C = 5$, so $C = 5$ and $f(x) = 2x^3 - 2x + 5$.

Flashcards

$displaystyleint x^n,dx$

$\dfrac{x^{n+1}}{n+1} + C$, for $n \ne -1$

$displaystyleint (ax + b)^n,dx$

$\dfrac{(ax + b)^{n+1}}{a(n + 1)} + C$

Finding $C$

Substitute a known point into the primitive.

Practice

$\displaystyle\int x^5\,dx = $
$\displaystyle\int (2x + 3)^4\,dx = $
$\displaystyle\int x^{-2}\,dx = $
$\displaystyle\int \sqrt{4x + 1}\,dx = $
Evaluate $\displaystyle\int_0^1 (2x + 1)^3\,dx$.
Evaluate $\displaystyle\int_0^2 (3x - 2)^2\,dx$.
$f'(x) = 6x^2 - 2$ and $f(1) = 5$. Find $f(0)$.
A curve has $\dfrac{dy}{dx} = 2x + 1$ and passes through $(1, 4)$. Find $y$ when $x = 2$.
Why does $\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ fail for $n = -1$?