Calculus I
&
Engineering Mathematics 2

Workshop 6


Integration

Recall that if $y=f(x),$ the area under the curve over the interval $I = [a,b]$ is

$\displaystyle \int_I f(x)dx = \lim_{n\to \infty} \sum_{i=1}^{n} f(x_i^*)(x_i-x_{i-1})$

where $x_i^* \in [x_i,x_{i-1}]$.




Integration

$\displaystyle \int_I f(x)dx = \lim_{n\to \infty} \sum_{i=1}^{n} f(x_i^*)(x_i-x_{i-1})$


Integration

Right, Middle, Left Riemann sums


Antiderivatives to compute Definite Integrals

Consider the continuous function $f(x)$ on $[a,b]$ and $F(x)$ is function such that $F'(x)=f(x)$ for every $x\in[a,b],$ then

• We call $F(x)$ an anti-derivative.

• This fact is known as The Fundamental Theorem of Calculus.



Example

$\ds \int_1^3 x^2 \, dx$ $\ds = \left[\dfrac{x^3}{3}+C\right]_1^3$ $\qquad \quad \quad\; F(x) = \dfrac{x^3}{3}+C\;$ 👈

$\qquad \qquad \ds = \underbrace{\left[\dfrac{(3)^3}{3}+C\right]}_{F(3)} - \underbrace{\left[\dfrac{(1)^3}{3}+C\right]}_{F(1)}$

$\qquad \qquad \ds = \dfrac{27}{3}+C $ $ -\, \dfrac{1}{3}-C$

$\qquad \qquad \ds = \dfrac{26}{3}$


Example

$\ds \int_1^3 x^2 \, dx$ $\ds = \left[\dfrac{x^3}{3}\right]_1^3$ $\qquad \quad \quad\; F(x) = \dfrac{x^3}{3}\;$ 👈

$\qquad \qquad \ds = \underbrace{\left[\dfrac{(3)^3}{3}\right]}_{F(3)} - \underbrace{\left[\dfrac{(1)^3}{3}\right]}_{F(1)}$

$\qquad \qquad \ds = \dfrac{27}{3} $ $ -\, \dfrac{1}{3}$

$\qquad \qquad \ds = \dfrac{26}{3}$

👉 Note we don't need
      to write the constant $C$.


Integration rules

Integration by substitution

\(\ds \int f\left(g\left(x\right)\right)\, dx \) \(\ds = \int f(u)\, du\,\) where \(\,u = g(x).\)

Integration by parts

\(\ds \int \frac{df\left(x\right)}{dx}g\left(x\right)\, dx \) \(\ds = f\left(x\right)g\left(x\right) \) \(\ds \, - \int f\left(x\right)\frac{dg\left(x\right)}{dx}\, du\)




Some Standard Indefinite Integrals
  • \(\displaystyle \int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C,\quad n \neq -1\)
  • \(\displaystyle \int \dfrac{1}{x}\,dx = \ln|x| + C\)
  • \(\displaystyle \int e^x\,dx = e^x + C\)
  • \(\displaystyle \int \sin x\,dx = -\cos x + C\)
  • \(\displaystyle \int \cos x\,dx = \sin x + C\)
  • \(\displaystyle \int \sec^2 x\,dx = \tan x + C\)
  • \(\displaystyle \int \dfrac{1}{1+x^2}\,dx = \arctan x + C\)
  • \(\displaystyle \int \dfrac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C\)
  • \(\displaystyle \int \sinh x\,dx = \cosh x + C\)
  • \(\displaystyle \int \cosh x\,dx = \sinh x + C\)
  • \(\displaystyle \int \sech^2 x\,dx = \tanh x + C\)
  • \(\displaystyle \int \csch^2 x\,dx = -\coth x + C\)
Integration by substitution \(\ds \int f\left(g\left(x\right)\right)\, dx = \int f(u)\, du,\,\) where \(\,u = g(x)\)
Integration by parts \(\ds \int \frac{df\left(x\right)}{dx}g\left(x\right)\, dx = f\left(x\right)g\left(x\right) - \int f\left(x\right)\frac{dg\left(x\right)}{dx}\, du\)

Credits