Integrate a function with respect to a different variable?

Question

I'm suppose to integrate this function,$$\int\left(e^{3t}-\frac{3}{t}\right)dx$$ I took the antiderivative and got the incorrect answer, the correct one being:

$$\left(e^{3t}-\frac{3}{t}\right)x$$

What did I do wrong?

Answer 1

First, try to solve $$\int 1\,dx \text{.}$$

Then solve $$\int a\,dx \text{,}$$ which is easy since it's just another way of writing $$a\int 1\,dx\text{.}$$

Now replace $a$ by your integrand $e^{3t} - \frac{3}{t}$. You may do that because it's value doesn't depend on $x$, so for the purpose of integrating over $x$, it's just a plain old constant.