Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 4

Table of contents
- a 4
  - Solution a
- b 4
  - Solution b

a

Solution

The fundamental theorem of calculus applies because the integrand is continuous. Thus

\frac{d I}{d x} = f (x) .

The requested derivative is

d I / d x

, as shown in the PDF; the local OCR corrupts its notation.

b

Solution

Use the product rule and the fundamental theorem of calculus:

\frac{d}{d x} [sin x \int_{0}^{x} sin y d y] = cos x \int_{0}^{x} sin y d y + sin^{2} x .

Since the integral is

1 - cos x

, this simplifies to

cos x - cos (2 x) .

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