Integrals

Calculus in the ancient times: Method of exhaustion (see wikipedia)

User-friendly definition of the integral:

lim_{N\to\infty}\frac{b-a}{N}\sum_{i=1}^N f(x_i)

It would be better if the integral would not depend on the definition of the bins. In the limit N\to\infty it doesn’t matter if they are evenly spaced or not:

lim_{N\to\infty}\sum_{i=1}^N f(x_i)dx_i

Fundamental theorem of calculus:

If f(x) is a continuous real-valued function on a closed interval [a,b]

F(x) = \int_a^x f(t) dt

then F(x) is continuous and differentiable in [a,b] and:

DF(x) = f(x)

as a corollary:

\int_a^b f(x) dx = F(b)-F(a)

Exercises

\int \left ( 3x^{-5} - 7x^3 + 3 - x^9\right ) dx

\int \sin^3x \cos x dx

\int{ \left ( 2x + 3\right ) \left ( x^2 + 3 x + 15 \right ) dx}

\int e^x e^{-3 x} dx

\int x^2e^x dx (tip: by parts)

Try to solve a more general formula: \int x^n e^x dx

(get the solution using Wolfram:alpha )

For the die-hard:

\int \frac{dx}{ \sqrt{3-2 x^2} }

( solution )

tip: make a substitution deplyoing this relationship:

\cos^2 \theta = 1 - \sin^2 \theta

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