Question

Evaluate the indefinite integral: ∫dx / (1 + x²) = [insert answer here] + C.

A. arctan(x) + C
B. ln(1 + x²) + C
C. sin(x) + C
D. e^(x²) + C

Answer 1

To evaluate the indefinite integral, we can use the trigonometric substitution method by substituting x = tan(θ). The final answer is arctan(x) + C.

Step-by-step explanation:

To evaluate the indefinite integral ∫dx / (1 + x²), we can use the trigonometric substitution method. Let's substitute x = tan(θ), which implies dx = dθ / cos²(θ).

Now, the integral becomes ∫(dθ / (1 + tan²(θ))) = ∫(dθ / sec²(θ)) = ∫cos²(θ)dθ.

Using the double angle identity for cosine, we have ∫(1 + cos(2θ))/2 dθ = (θ + (sin(2θ))/2) + C.

Substituting back x = tan(θ), the final answer is arctan(x) + C. Therefore, the correct option is A. arctan(x) + C.