Definite Integral

Question 1

How to calculate a definite integral over an interval?

Answer

To perform an integral calculation (integration), first, calculate the corresponding primitive function.

For a function $ f(x) $ to be integrated over $ [a;b] $ and $ F(x) $ the primitive of $ f(x) $. Then $$ \int^b_a f(x) \mathrm{ dx} = F(b)-F(a) $$

Example: Integrate $ f(x) = x $ over the interval $ [0;1] $. Calculate its primitive $ F(x) = \frac{1}{2} x^2 $ and so integral $$ \int^1_0 f(x) \mathrm {dx} = F(1) - F(0) = \frac{1}{2} $$

Enter the function, its lower and upper bounds and the variable to integrate, dCode will make the computation.

Question 2

What is the list of common primitives?

Answer

Function	Primitive
$$ \int \,\rm dx$$	$$x + C$$
$$ \int x^n\,\rm dx$$	$$ \frac{x^{n+1}}{n+1} + C \qquad n \ne -1 $$
$$ \int \frac{1}{x}\,\rm dx$$	$$\ln \left\| x \right\| + C \qquad x \ne 0 $$
$$ \int \frac{1}{x-a} \, \rm dx $$	$$\ln \| x-a \| + C \qquad x \ne a $$
$$ \int \frac{1}{(x-a)^n} \, \rm dx$$	$$-\frac{1}{(n-1)(x-a)^{n-1}} + C \qquad n \ne 1 , x \ne a $$
$$ \int \frac{1}{1+x^2} \, \rm dx$$	$$\operatorname{arctan}(x) + C $$
$$ \int \frac{1}{a^2+x^2} \, \rm dx$$	$$\frac{1}{a}\operatorname{arctan}{ \left( \frac{x}{a} \right) } + C \qquad a \ne 0 $$
$$ \int \frac{1}{1-x^2} \, \rm dx$$	$$\frac{1}{2} \ln { \left\| \frac{x+1}{x-1} \right\| } + C $$
$$ \int \ln (x)\,\rm dx$$	$$x \ln (x) - x + C $$
$$ \int \log_b (x)\,\rm dx$$	$$x \log_b (x) - x \log_b (e) + C $$
$$ \int e^x\,\rm dx$$	$$e^x + C $$
$$ \int a^x\,\rm dx$$	$$\frac{a^x}{\ln (a)} + C \qquad a > 0 , a \ne 1 $$
$$ \int {1 \over \sqrt{1-x^2}} \, \rm dx$$	$$\operatorname{arcsin} (x) + C $$
$$ \int {-1 \over \sqrt{1-x^2}} \, \rm dx$$	$$\operatorname{arccos} (x) + C $$
$$ \int {x \over \sqrt{x^2-1}} \, \rm dx$$	$$\sqrt{x^2-1} + C $$
$$ \int \sin(x)\,\rm dx $$	$$ -\cos(x)+C $$
$$ \int \cos(x)\,\rm dx $$	$$ \sin(x)+C $$
$$ \int \tan(x)\,\rm dx $$	$$ -\ln\|\cos(x)\|+C $$

Reminder: the derivative of the primitive of a function is the function itself.

Question 3

What are the functions E, F, I0, K0?

Answer

Calculation of some forms of integrals involve special functions such as $ E $ and $ F $ which are elliptic integrals or $ I_0, I_n, J_0, J_n, K_0, K_n $ which are Bessel functions.

Definite Integral

Primitive Calculator