In[1]:= Integrate[x^2 Exp[x*y], {x, 0, 1}, {y, 0, 2}]
Out[1]= 1/4 (-1 + E^2)
In[2]:= i1 = Integrate[x^2 Exp[x*y], {y, 0, 2}]
Out[2]= (-1 + E^(2 x)) x
In[3]:= Integrate[i1, {x, 0, 1}]
Out[3]= 1/4 (-1 + E^2)
The only hard integral is x*exp(2*x) which Wolfram Alpha can show step by step:
Take the integral:
integral e^(2 x) x dx
For the integrand e^(2 x) x, integrate by parts, integral f dg = f g- integral g df, where
f = x, dg = e^(2 x) dx,
df = dx, g = e^(2 x)/2:
= 1/2 e^(2 x) x-1/2 integral e^(2 x) dx
For the integrand e^(2 x), substitute u = 2 x and du = 2 dx:
= 1/2 e^(2 x) x-1/4 integral e^u du
The integral of e^u is e^u:
= 1/2 e^(2 x) x-e^u/4+constant
Substitute back for u = 2 x:
= 1/2 e^(2 x) x-e^(2 x)/4+constant
Which is equal to:
Answer: |
| = 1/4 e^(2 x) (2 x-1)+constant
I wasn't joking when I said "Integrate by Parts"