Exercises

1. Write down the value of each of the following integrals. You should be able to do this without any calculation.

   \(\displaystyle \int_{-\pi}^{\,\pi} \cos\,5t \,dt\)	\(\displaystyle \int_{-\pi}^{\,\pi} \sin^2\,t \,dt\)	\(\displaystyle \int_{-\pi}^{\,\pi} \cos\,4t \,\cos\,6t\,dt\)
   \(\displaystyle \int_{-\pi}^{\,\pi} \cos^2\,5t \,dt\)	\(\displaystyle \int_{-\pi}^{\,\pi} \sin\,2t \,\cos\,7t\,dt\)	\(\displaystyle \int_{-\pi}^{\,\pi} \sin\,3t \,\sin\,5t\,dt\)

2. Calculate each of the following integrals.
   

   a.   \(\displaystyle \int_{-3}^{\,3}\, 1\,dt\)	b.   \(\displaystyle \int_{-3}^{\,3}\, \sin\,\frac{\pi}{3} t\,dt\)	c.   \(\displaystyle \int_{-3}^{\,3} \,\cos\,\frac{\pi}{3} t\,dt\)
   d.   \(\displaystyle \int_{-3}^{\,3} \,\cos\,\frac{2 \pi}{3} t\,dt\)	e.   \(\displaystyle \int_{-3}^{\,3} \,\sin^2\,\frac{2 \pi}{3} t\,dt\)	f.   \(\displaystyle \int_{-3}^{\,3} \,\cos^2\,\frac{2 \pi}{3} t\,dt\)

3. Assume \(f\) is a function of the form

   \(\displaystyle f(t)=b_0+a_1\,\sin\, \frac{\pi}{3} t+b_1\,\cos\,\frac{\pi}{3} t+b_2\,\cos\,\frac{2\pi}{3} t\).

   Find a formula for \(f\) if

   \(\displaystyle \int_{-3}^{\,3}\,f(t)\,dt=6\), \(\displaystyle \int_{-3}^{\,3}\,f(t)\,\cos\,\frac{\pi}{3} t\,dt=2\),
   \(\displaystyle \int_{-3}^{\,3}\,f(t)\,\sin\,\frac{\pi}{3} t\, dt=-3\), \(\displaystyle \int_{-3}^{\,3}\,f(t)\,\cos\,\frac{2 \pi}{3} t\,dt=9\).

4. Assume \(f\) is a function of the form

   \(\displaystyle f(t)=b_0+a_1\,\sin\, \frac{\pi}{5} t+b_1\,\cos\,\frac{\pi}{5} t+b_2\,\cos\, \frac{2\pi}{5} t\).

   Find a formula for \(f\) if

   \(\displaystyle \int_{-5}^{\,5}\,f(t)\,dt=5\), \(\displaystyle \int_{-5}^{\,5}\,f(t)\,\cos\,\frac{\pi}{5} t\,dt=4\),
   \(\displaystyle \int_{-5}^{\,5}\,f(t)\,\sin\,\frac{\pi}{5} t \,dt=-5\), \(\displaystyle \int_{-5}^{\,5}\,f(t)\,\cos\, \frac{2 \pi}{5} t\,dt=20\).