Calculate the following indefinite integrals.

[1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$

[2] $\int x\log_{10} x dx$

[3] $\int \frac{x}{\sqrt{2x-1}}dx$

[4] $\int (x^2+1)\ln x dx$

[5] $\int e^x\cos x dx$
### Problem 1: 
Evaluate the integral $\int \frac{\sin x \cos x}{1 + \sin^2 x} \, dx$.

1. Let $u = \sin x$. Then $du = \cos x \, dx$.
2. Substitute $u$ and $du$ into the integral:
   \[
   \int \frac{\sin x \cos x}{1 + \sin^2 x} \, dx = \int \frac{u \, du}{1 + u^2}
   \]
3. Recognize that $\int \frac{u}{1 + u^2} \, du$ can be simplified using the substitution $v = 1 + u^2$, but a simpler approach is to notice that the numerator is the derivative of the denominator:
   \[
   \int \frac{u}{1 + u^2} \, du = \frac{1}{2} \int \frac{2u}{1 + u^2} \, du = \frac{1}{2} \ln |1 + u^2| + C
   \]
4. Substitute back $u = \sin x$:
   \[
   \frac{1}{2} \ln |1 + \sin^2 x| + C
   \]

The final answer is $\boxed{\frac{e^x (\sin x + \cos x)}{2} + C}$