Basic Integration Formulas [Rumus Dasar Integral]

When solving a differential equation of the form $\dfrac{dy}{dx}=f(x)$
it is convenient to write it in the equivalent differential form $dy=f(x)dx$

The operation of finding all solutions of this equation is called antidifferetiation [0r indefinite integration] adn is denoted by an integral sign $\int$.

The general solution is denoted by $y=\int f(x)dx=F(x)+C$
\begin{split}
f(x)&= integrand\\
d(x)&= Variable\ of\ integration\\
F(x)&=An\ antiderivative\ of\ f(x)\\
C&=Constant\ of\ integration
\end{split}
The expression $\int f(x)dx$ is read as the antiderivative of $f$ with respect to $x$. So, the differential $dx$ serves to identify $x$ as the variable of integration. The term indefinite integral is a synonym for antiderivative.

Basic Integration Rules

The inverse nature of integration and differentiation can be verified by substituting $F'(x)$ for $f(x)$ in the indefinite integration definotion to obtain;
$\int F'(x)dx=F(x)+c$

Moreover, if $\int f(x)dx=F(x)+C$, then
$\dfrac{d}{dx}\left [ \int f(x)dx) \right ]=f(x)$
  1. $\int x^{n}dx=\dfrac{1}{n+1}x^{n}+C,\ n\neq -1$
  2. $\int kf(x)dx=k\int f(x)dx$
  3. $\int [f(x) \pm g(x)]dx=\int f(x)dx \pm \int g(x)dx$
  4. $\int dx= x + C$
  5. $\int a^{x} dx= \left (\dfrac{1}{ln\ a} \right )a^{x} + C$
  6. $\int e^{x} dx= e^{x} + C$
  7. $\int sin\ x\ dx= -cos\ x + C$
  8. $\int sin\ u\ dx= -\dfrac{1}{u'}cos\ u + C$
  9. $\int cos\ x\ dx= sin\ x + C$
  10. $\int cos\ u\ dx= \dfrac{1}{u'}sin\ u + C$
  11. $\int tan\ x\ dx= -ln|cos\ x| + C$
  12. $\int cot\ x\ dx= ln|sin\ x| + C$
  13. $\int sec\ x\ dx= ln|sec\ x+tan\ x| + C$
  14. $\int csc\ x\ dx= -ln|csc\ x+cot\ x| + C$
  15. $\int sec^{2}\ x\ dx= tan\ x + C$
  16. $\int sec^{2}\ u\ dx=\dfrac{1}{u'} tan\ u + C$
  17. $\int csc^{2}\ x\ dx= -cot\ x + C$
  18. $\int csc^{2}\ u\ dx= -\dfrac{1}{u'}cot\ u + C$
  19. $\int sec\ x\ tan\ x\ dx= sec\ x + C$
  20. $\int csc\ x\ cot\ x\ dx= -csc\ x + C$
  21. $\int \dfrac{dx}{\sqrt{a^{2}-x^{2}}}=arcsin\dfrac{u}{a}+C$
  22. $\int \dfrac{dx}{a^{2}+x^{2}}=\dfrac{1}{a}arctan\dfrac{u}{a}+C$
  23. $\int \dfrac{dx}{x\sqrt{x^{2}-a^{2}}}=\dfrac{1}{a}arcsec\dfrac{|u|}{a}+C$

Partial Integration

$\int u\ dv=u \cdot v-\int v\ du$

The Fundamental Theorem Of Calculus

If a function $f$ is continuous on the closed interval $[a,b]$ and $F$ is an antiderivative of $f$ on the interval $[a,b]$, then
$\int_{a}^{b}f(x)dx=F(b)-F(a)$

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