Mathematics

This is a vector integral, where you integrate each component separately. The key insight is that each component is a standard antiderivative in disguise: $\frac{1}{t+1}$ gives a logarithm, $\frac{1}{t^2+1}$ is arctangent, and $\frac{t}{t^2+1}$ is a substitution away from a logarithm. Evaluate the antiderivative of each component at the bounds and combine the results as a vector.

This is a vector integral, where you integrate each component separately. The key insight is that each component is a standard antiderivative in disguise: $\frac{1}{t+1}$ gives a logarithm, $\frac{1}{t^2+1}$ is arctangent, and $\frac{t}{t^2+1}$ is a substitution away from a logarithm. Evaluate the antiderivative of each component at the bounds and combine the results as a vector.