Answer:
Sure, here's how to solve the integral ∫x/(x+1) dx using substitution:
1. Let u = x + 1, which means that du/dx = 1.
2. Solve for x in terms of u: x = u - 1.
3. Rewrite the integral in terms of u:
∫x/(x+1) dx = ∫(u-1)/u du
4. Simplify the integrand by expanding the numerator:
∫(u-1)/u du = ∫(u/u) - (1/u) du
5. Integrate each term separately:
∫(u/u) - (1/u) du = ∫1 - (1/u) du
= u - ln|u| + C, where C is the constant of integration.
6. Substitute back in for u using the substitution we made earlier:
u = x + 1
= x + 1 - ln|x + 1| + C
So the solution to the integral ∫x/(x+1) dx using substitution is x + 1 - ln|x + 1| + C, where C is the constant of integration.