Can you spot the mistake?

Question

Let $$I=\displaystyle\int\sin^2x \ dx$$then

$$I=\displaystyle\int(1-\cos^2x) \ dx=x+C-\displaystyle\int \cos^2x \ dx$$

Using the substitution $x=x+\frac{\pi}{2}$ we get

$$I=x+C-\displaystyle\int\cos^2(x+\frac{\pi}{2}) \ dx=x+C-\displaystyle\int\sin^2x \ dx$$

and thus $I=\frac{x}{2}+C$ which is obviously wrong.

So where's the mistake in the above argument??

Answer 1

You cannot say that $x=x+\pi/2$. This makes no sense mathematically.

It would be better to use $$\sin^2 x = \frac{1-\cos(2x)}{2}$$

Result: $$I=\frac{x}{2}-\frac{\sin(2x)}{4}+C$$

Answer 2

If you put $u=x+ {\pi\over2}$,then you know that you will need to replace u with x.

Yet you used $x$, and forgot to do so.

Answer 3

Substitution does not work this way.

Even simpler functions can be used to illustrate the point:

Let $$J=\int x \,dx$$
Then letting $x=x+1$ we see $$\int x\,dx =\int (x+1)\,dx=\int x\,dx + \int 1 \,dx\implies \int 1\,dx = C$$

But the fact is that $x\neq x+1$, so this sort of substitution was not valid.

True, in working with definite integrals, people are often careless and re-use the same variable name. So long as you are careful to change the limits of integration along with the variable, there is no real harm in that (other than that it can be a bit confusing).