Section 4.4 Indefinite Integrals

If $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$,

  • the definite integral $\displaystyle \int_a^b f(x) \, dx = F(b) - F(a)$ is a number. We find the definite exactly value.
  • the indefinite integral is a function or a family of functions. $$ \int f(x) \, dx = F(x) + C $$ where $C$ is some constant.

Why $+C$ ?

Consider the following derivatives:

  • $\displaystyle \dfrac{d}{dx} \left( \frac{x^3}{3} + 1 \right) = x^2$
  • $\displaystyle \dfrac{d}{dx} \left( \frac{x^3}{3} - 7 \right) = x^2$
  • $\displaystyle \dfrac{d}{dx} \left( \frac{x^3}{3} + 42 \right) = x^2$
  • $\displaystyle \dfrac{d}{dx} \left( \frac{x^3}{3} + C \right) = x^2$ where $C$ is a constant.

…

since the derivative of that family of functions is always $x^2$, any of them is an antiderivative of $x^2$.

Many, many integrals

  • $\displaystyle\int cf(x)\,dx=c \int f(x)\, dx$
  • $\displaystyle\int(f(x) + g(x))\, dx=\int f(x)\, dx + \int g(x)\, dx $
  • $\displaystyle\int k \,dx=kx + C$
  • $\displaystyle\int x^n \, dx =\dfrac{x^{n+1}}{n+1} + C$ if $n \ne -1$
  • $\displaystyle\int \cos x \, dx =\sin x +C $
  • $\displaystyle\int \sin x \,dx =-\cos x +C $
  • $\displaystyle\int \sec^2 x \, dx=\tan x +C $
  • $\displaystyle\int \sec x \tan x\, dx=\sec x +C $

Any time you integrate, you can always check your answer by differentiating!

Example

Evaluate the integral $$ \int 15x^4 - \cos x \, dx $$

Example

Evaluate the integral $$ \int_0^2 x^3 - 5x \, dx $$

Example

Find the area under the $y=\sin x$ between $-\pi/2$ and $\pi/2$

Example

Evaluate the integral $$ \int \sec t (\sec t + \tan t) \, dt $$