Answer: t = tan(x/2) (Weierstrass substitution).
- A u = cos x, reducing the integrand to a rational function of u directly
- B t = tan(x/2) (Weierstrass substitution)
- C u = sin x, used for integrands containing only odd powers of cosine
- D No substitution needed since the integral has an elementary antiderivative
Correct answer: B. t = tan(x/2) (Weierstrass substitution)
Explanation: Weierstrass substitution t=tan(x/2): cosx=(1−t²)/(1+t²), dx=2dt/(1+t²). Denominator a+b·(1−t²)/(1+t²) becomes rational in t, converting the trigonometric integral into a standard rational integral.
The exact area under the curve from a to b (shaded) is approximated by a few rectangles, the Riemann sum idea behind the definite integral.