When finding the derivative of the integral of a function I get different results

Question

I am using wolframalpha.com and other online calculus calculators. The problem I am solving:

$$\int x\cos^2\left(x^2\right)dx$$

But the integral and its derivative don't match. Am I doing something wrong? What am I typing in wrong? I think it is something as simple as a mistype but I cannot see where.

Answer 1

Infact the derivative of the integral is just the same as $x\cos^2(x^2)$. You just have to use the identity: $$\cos 2x +1= 2\cos^2x$$ Replacing $x$ by $x^2$, we have$$\cos(2x^2)+1=2\cos^2(x^2)$$ Hence we have $$\begin{align} \frac12x(\cos (2x^2)+1)&=\frac12x(2\cos^2(x^2))\\ &=x\cos^2(x^2) \end{align}$$

Answer 2

If we have:

$$\int f(x) dx= g(x)$$ and $$\int f(x) dx = h(x)$$

This implies that $g(x)=h(x)+C$ for every $x$.

Therefore, especially in the integrals of trigonometric functions, it is very common to achieve two seemingly different integrals of the same function, but it's usually quite simple to show equivalence.

In this case, as Faiq's answer points out, the two expressions are identical.