Question

∫x/(x+1) dx using substitution. step by step method

Answer 1

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.