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Integral Calculus

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 INTEGRATION

What is Integration?

Integration is the reverse process to that of differentiation. By differentiation we find the derivative of the given function, whereas by integration we find the function whose derivative is known. This function is called the integral of the given function. Integration is sometimes also referred to as indefinite integral.

GENERAL FORMULA

PROPERTIES OF INDEFINITE INTEGRALS


If f (x) is defined and continuous on [a, b], except maybe at a finite number of points, then we have

  • $\displaystyle \int_{c}^{c}$f (x) dx = 0;
  • $\displaystyle \int_{a}^{b}$f (x) dx = $\displaystyle \int_{a}^{c}$f (x) dx + $\displaystyle \int_{c}^{b}$f (x) dx;
  • $\displaystyle \int_{b}^{a}$f (x) dx = - $\displaystyle \int_{a}^{b}$f (x) dx;

for any arbitrary numbers a and b, and any c $ \in$ [a, b].

COMMON SUBSTITUTIONS

1.
$\displaystyle \int F(ax+b)dx = \displaystyle \frac{1}{a} \displaystyle \int F(u)du$
where $u=ax\,+\,b$
2.
$\displaystyle \int F\left(\displaystyle\sqrt{ax\, +\, b}\right)\, dx = \displaystyle \frac{2}{a} \displaystyle \int u\,F(u)\,du $
where $u=\displaystyle\sqrt{ax\,+\,b}$
3.
$\displaystyle \int F\left( \sqrt[n]{ax+b} \right) \,dx = \displaystyle \frac{n}{a} \displaystyle \int u^{n-1}\,F(u)\,du$
where $u=\sqrt[n]{ax+b}$
4.
$\displaystyle \int F\left( \displaystyle\sqrt{a^{2}-x^{2}}\right)\,dx = a\,\displaystyle \int F(a \cos u)\,\cos u\,du$
where $x=a\sin u$
5.
$\displaystyle \int F\left( \displaystyle\sqrt{x^2+a^{2}} \right)\,dx= a\,\displaystyle \int F(a \sec u) \sec ^{2} u \, du$
where $x=a\tan u$
6.
$\displaystyle \int F\left( \displaystyle\sqrt{x^{2}-a^{2}} \right)\,dx=a \displaystyle \int F(a\tan u) \sec u \tan u\,du$
where $x=a\sec u$
7.
$\displaystyle \int F (e\displaystyle^{ax})\,dx = \displaystyle \frac{1}{a} \displaystyle \int\displaystyle \frac{F(u)}{u}\,du$
where $u=e\displaystyle^{ax}$
8.
$\displaystyle \int F(\ln x)\,dx = \displaystyle \int F(u)\,e\displaystyle^u\,du$
where $u=\ln x$
9.
$\displaystyle \int F\left( \sin ^{-1}\displaystyle \frac{x}{a}\right)\,dx = a\,\displaystyle \int F(u)\cos u\,du$
where $u=\sin ^{-1}\displaystyle \frac{x}{a}$
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