A-Level Mathematics May/June 2025 Q3: Find the exact value of ∫(from -π/4 to π/4) 3 cos² 5x dx.

1 answer

• ▲0
  accepted ✓

  The problem is to find the exact value of the definite integral $\int_{-\pi/4}^{\pi/4} 3 \cos^2(5x) \,dx$.

  First, we must use a trigonometric identity to express $\cos^2(5x)$ in a form that can be integrated. We use the double angle identity $\cos(2A) = 2\cos^2(A) - 1$, which rearranges to $\cos^2(A) = \frac{1}{2}(1 + \cos(2A))$.

  Letting $A = 5x$, we get $\cos^2(5x) = \frac{1}{2}(1 + \cos(10x))$.

  The integral becomes: $\int_{-\pi/4}^{\pi/4} 3 \left( \frac{1}{2}(1 + \cos(10x)) \right) \,dx = \frac{3}{2} \int_{-\pi/4}^{\pi/4} (1 + \cos(10x)) \,dx$

  Now, we perform the integration: $\frac{3}{2} \left[ x + \frac{1}{10}\sin(10x) \right]_{-\pi/4}^{\pi/4}$ $= \left[ \frac{3}{2}x + \frac{3}{20}\sin(10x) \right]_{-\pi/4}^{\pi/4}$

  Next, we substitute the upper and lower limits and subtract: $= \left( \frac{3}{2}\left(\frac{\pi}{4}\right) + \frac{3}{20}\sin\left(10 \cdot \frac{\pi}{4}\right) \right) - \left( \frac{3}{2}\left(-\frac{\pi}{4}\right) + \frac{3}{20}\sin\left(10 \cdot -\frac{\pi}{4}\right) \right)$ $= \left( \frac{3\pi}{8} + \frac{3}{20}\sin\left(\frac{5\pi}{2}\right) \right) - \left( -\frac{3\pi}{8} + \frac{3}{20}\sin\left(-\frac{5\pi}{2}\right) \right)$

  We evaluate the trigonometric terms. Since $\sin(\frac{5\pi}{2}) = \sin(\frac{\pi}{2} + 2\pi) = \sin(\frac{\pi}{2}) = 1$, and $\sin(-\theta) = -\sin(\theta)$, we have $\sin(-\frac{5\pi}{2}) = -1$. $= \left( \frac{3\pi}{8} + \frac{3}{20}(1) \right) - \left( -\frac{3\pi}{8} + \frac{3}{20}(-1) \right)$ $= \left( \frac{3\pi}{8} + \frac{3}{20} \right) - \left( -\frac{3\pi}{8} - \frac{3}{20} \right)$ $= \frac{3\pi}{8} + \frac{3}{20} + \frac{3\pi}{8} + \frac{3}{20}$ $= 2\left(\frac{3\pi}{8}\right) + 2\left(\frac{3}{20}\right)$ $= \frac{6\pi}{8} + \frac{6}{20}$

  Simplifying the fractions gives the final exact answer: $= \frac{3\pi}{4} + \frac{3}{10}$

  ---

  How the marks are awarded

  • M1 — The first mark is for correctly applying the double angle identity for cosine, $\cos^2(A) = \frac{1}{2}(1 + \cos(2A))$, to transform the integrand from $3\cos^2(5x)$ into $\frac{3}{2}(1 + \cos(10x))$.
  • A1 — The second mark is awarded for correctly integrating the transformed expression. Integrating $\frac{3}{2}(1 + \cos(10x))$ with respect to $x$ gives the correct result of $\frac{3}{2}x + \frac{3}{20}\sin(10x)$.
  • DM1 — This dependent method mark is for substituting the limits, $\frac{\pi}{4}$ and $-\frac{\pi}{4}$, correctly into the integrated expression, as shown by the line $(\frac{3\pi}{8} + \frac{3}{20}\sin(\frac{5\pi}{2})) - (-\frac{3\pi}{8} + \frac{3}{20}\sin(-\frac{5\pi}{2}))$.
  • A1 — The final accuracy mark is for obtaining the correct final answer in its exact, simplified form, which is $\frac{3\pi}{4} + \frac{3}{10}$.

  Common mistakes

  • Using the incorrect double angle formula, such as $\cos^2(5x) = \frac{1}{2}(1 - \cos(10x))$ or forgetting to double the angle to get $10x$.
  • Making an error when integrating $\cos(10x)$, for example forgetting the $\frac{1}{10}$ factor to get $\sin(10x)$ instead of $\frac{1}{10}\sin(10x)$.
  • Incorrectly evaluating the trigonometric functions at the limits, for example calculating $\sin(\frac{5\pi}{2})$ as 0 or -1, or making a sign error with $\sin(-\frac{5\pi}{2})$.
  • Making an arithmetic error when substituting the limits, such as incorrectly subtracting the negative terms, for example $(\frac{3\pi}{8}) - (-\frac{3\pi}{8}) = 0$.

  Examiner tip: Recognise that powers of trigonometric functions like $\cos^2(x)$ or $\sin^2(x)$ usually require a double angle formula to be integrated.

  ---

  AI-generated model answer, grounded in the official Cambridge mark scheme and reviewed by the MarkScheme team. Mark your own answer to this question →

  @markscheme_answers

Your answer

Sign in to answer this question.