Final answer:
The substitution method used is U-substitution, where u = x+1, leading to an integral that results in arctan(x+1) + C.
Step-by-step explanation:
U-substitution The substitution method used to evaluate the integral ∫ 1/(x+1)²+1 dx is C) U-substitution. This technique involves substituting a part of the integral with a new variable to simplify the integration process. For this integral, we can set u = x+1, which then implies du = dx. This substitution simplifies the integral to ∫ 1/u²+1 du, which is a standard form that can be integrated directly to arctan(u) + C, where C is the integration constant. After substituting back for x, we write the final answer as arctan(x+1) + C.
The substitution method used to evaluate the integral ∫ 1/₍ₓ₊1₎²₊1 dx is U-substitution.To solve this integral using U-substitution, we let u = x + 1, which means du = dx. Substituting these values into the integral, we get ∫ 1/u² du. Now, we can evaluate this integral as -1/u + C = -1/(x + 1) + C.Therefore, the correct answer is option C) U-substitution.