Can I merge a term multiplied to a bracket raised to a power with that bracket?

Question

To make my point clearer: consider the integral $$\int\ e^{2x}(x+1)^2 dx$$ one way to solve this is by parts. But a friend suggested something else that seemed to work. He combined the term and the bracket to form $$\int\ (xe^x+e^x)^2 dx$$ which is a basic integral that can be solved easily. He asked a teacher and the teacher called it stupid and told him to never use it again. That's the example he showed the teacher: $$\int\ x^5(x^2+3)^4 dx = \int\ (x^{13/4} + 3x^{5/4})^4dx$$ Sorry if this sounds stupid. We just want to know what's right and what's wrong, and why.

Answer 1

For what concerns the first case, there is nothing wrong: indeed

$$(x^2 e^{2x} + e^{2x} + 2x e^{2x}) = (xe^x + e^x)^2$$

For the second, notice that

$$(x^2 + 3)^4 = x^8+12 x^6+54 x^4+108 x^2+81$$

and

$$(x^{13/4} + 3x^{5/4})^4 = x^8+12 x^6+54 x^4+108 x^2+81$$

So it holds.

Hence:

• Your friend is not stupid. Stupid is the professor calling him stupid

• The method is valid, there are no maths errors

• Despite that, the problem is that it's not very helpful. But not wrong. Just unhelpful.

Example

Take the first one: with a simple substitution $x = t-1$ the integral becomes

$$e^{-2}\int e^{2t} t^2\ \text{d}t = \ \ \ \frac{e^{-2}}{4} e^{2 t} \left(2 t^2-2 t+1\right)$$

Where as integrating in your friend's way make the process harder.