Q.59 Calculate the following integral
∫0π2/4 sin√x dx = ________.
Solving ∫ from 0 to π2/4 of sin(√x) dx
The definite integral ∫0π2/4 sin(√x) dx equals 2. This result comes from a substitution that simplifies the integrand into a standard trigonometric integral. Use the substitution t = √x, so x = t2 and dx = 2t dt.
Step-by-Step Solution
Start with the substitution t = √x. When x = 0, t = 0; when x = π2/4, t = π/2. The differential dx = 2t dt transforms the integral.
The integral becomes ∫ from 0 to π/2 of sin(t) · 2t / t dt = ∫0π/2 2 sin(t) dt, since t cancels with the t in dx.
The antiderivative of 2 sin(t) is -2 cos(t). Evaluating from 0 to π/2 gives -2 cos(π/2) – (-2 cos(0)) = -2(0) – (-2(1)) = 2.
Common Options Explained
Multiple-choice options for this integral often include 1, π/4, π2/8, and 2. Option 1 arises from mistakenly evaluating ∫ sin(x) dx from 0 to π/2 without substitution.
π/4 or π2/8 might result from confusing limits or treating the integrand as sin(x)/√x instead of sin(√x).
The correct choice is 2, as verified by the substitution method across sources.
