1) Integrate (e^x+x)^2(e^x+1) dx
2) Integrate xe^x2 dx
Let u=(e^x+x)
du=(e^x +1) dx
I will be happy to critique your work or thinking. You are posting work for me to do, and I am not inclined to do that, it will not help you for me to do it. I think I gave you strong hints on a previous post: what is the problem with those?
To integrate the given expressions, we will have to make use of the substitution method. Let's go through each question step by step:
1) Integrate (e^x+x)^2(e^x+1) dx
Let u = (e^x + x).
To find du, we differentiate both sides of the equation with respect to x:
du = (e^x + 1) dx.
Now we can rewrite the given expression in terms of u:
(e^x + x)^2(e^x + 1) dx = u^2 du.
The integral now becomes:
∫ u^2 du.
Integrating u^2 du gives us:
∫ u^2 du = (1/3) u^3 + C.
Finally, substitute u back in terms of x:
(1/3) (e^x + x)^3 + C.
2) Integrate xe^x^2 dx
Let u = x^2.
To find du, we differentiate both sides of the equation with respect to x:
du = 2x dx.
Now we can rewrite the given expression in terms of u:
xe^x^2 dx = (1/2) e^u du.
The integral now becomes:
(1/2) ∫ e^u du.
Integrating e^u du gives us:
(1/2) e^u + C.
Finally, substitute u back in terms of x:
(1/2) e^(x^2) + C.
Remember, always check your integration by differentiating the result to verify if it matches the original expression!