Integral Calculus on Mathematica (2024)

MATHEMATICA can solve just about any integral as long as it can be expressed in terms of standard mathematical formulas.MATHEMATICA can solve indefinite, multiple, definite, and multiple definite integrals.

• Indefinite Integral
• Indefinite Multiple Integral
• Definite Integral
• Definite Multiple Integral

Indefinite Integral

To integrate an indefinite integral, you use the Integrate command. The Integrate command will be used in the following form:

Integrate[function,variable]

Lets consider an example. Lets integrate the function, x³+e^3x. Type:

Integrate[x^3+Exp[3x],x]

Hit the shift and return keys at the same time. This should yield:

 In[1]:=[x^3+Exp[3x],x]
Out[1]=E3x/3 + x4/4

Indefinite Multiple Integral

To integrate an indefinite multiple integral, you must use the Integrate command in the following fashion:

Integrate[function, variable 1, variable 2]

Lets consider the function, y⁴ + 2x^2y. Type the following:

Integrate[y^4+2x^2y,x,y]

Hit the shift and return keys at the same time. The output will yield:

 In[1]:=Integrate[y^4+2x^2y,x,y]
Out[1]=x3y3/3 + xy5/5

Definite Integral

The definite integral command takes the form:

Integrate[function,{variable,xmin,xmax}]

Lets consider the equation, x⁴+Sin 2x, and we want to evaluate the function on the boundary conditions, 0 < x < 3.14. Type the following:

Integrate[x^4+Sin[2x],{x,0,3.14}]

Hit the shift and return keys at the same time. The output will yield:

 In[1]:=Integrate[x^4+Sin[2x],{x,0,3.14}]
Out[1]=61.049

Definite Multiple Integral

The Integrate command will take the form:

Integrate[function,{variable,xmin,xmax},{variable,ymin,ymax}]

Lets consider the equation, 2xy, and the boundary conditions for this equation are, 0 < x < 5 and 0 < y < 2. Type the following:

Integrate[2xy,{x,0,5},{y,0,2}]

Hit the shift and return keys at the same time. The output should yield:

 In[1]:=Integrate[2xy,{x,0,5},{y,0,2}]
Out[1]=50