Integrals of Periodic Functions

  1. What is the area under a single “hump” of the graph of sine?
  2. The graphs of \(\sin(x)\) and \(\cos(x)\) bound infinitely many congruent copies of the same shape. What is the area of this shape?
  3. The graph of the function \(f(x) = \cos(x)\mathrm{e}^{\sin(x)}\) and the \(x\)-axis bound infinitely many congruent copies of the same shape. What is the area of this shape?

  4. Write down a formula for the indefinite integral of each of the following functions. Note that you can check your antiderivative using any software capable of symbolic integration, e.g. WolframAlpha.

    \(\displaystyle \int \cos(x) \,\mathrm{d}x \)
    \(\displaystyle \int \frac{1}{x} + 3\sin(x) \,\mathrm{d}x \)
    \(\displaystyle \int \sin(2x) \,\mathrm{d}x \)
    \(\displaystyle \int \cos(x-17) \,\mathrm{d}x \)
    \(\displaystyle \int \cos(3x+5) \,\mathrm{d}x \)
    \(\displaystyle \int (2x-1)\cos\bigl(x^2-x-2\bigr) \,\mathrm{d}x \)
    \(\displaystyle \int 3\sin(x)+x^2-1 \,\mathrm{d}x \)
    \(\displaystyle \int x\cos(x) \,\mathrm{d}x \)
    \(\displaystyle \int x^2\cos(x) \,\mathrm{d}x \)
    \(\displaystyle \int x^2\cos\bigl(x^3\bigr) \,\mathrm{d}x \)
    \(\displaystyle \int \sin(x)\cos(x) \,\mathrm{d}x \)
    \(\displaystyle \int \sqrt{\sin(x)}\cos(x)\,\mathrm{d}x \)
    \(\displaystyle \int \frac{\sin(x)}{\cos(x)} \,\mathrm{d}x \)
    \(\displaystyle \int \bigl(\cos(x)+1\bigr)\bigl(\sin(x)+1\bigr) \,\mathrm{d}x \)
    \(\displaystyle \int \sin(x)\bigl(\cos(x)\bigr)^9 \,\mathrm{d}x \)
    \(\displaystyle \int \bigl(\sin(x)\bigr)^2\bigl(\cos(x)\bigr)^9 \,\mathrm{d}x \)
    \(\displaystyle \int \sin(x)-\cos(x) \,\mathrm{d}x \)
    \(\displaystyle \int \frac{\sin\bigl(\sqrt{x}\bigr)}{\sqrt{x}} \,\mathrm{d}x \)
    \(\displaystyle \int \frac{3\cos\bigl(\tfrac{2}{x}\bigr)}{x^2}\,\mathrm{d}x \)
    \(\displaystyle \int -\tfrac{1}{x}\sin\bigl(\ln(x)\bigr) \,\mathrm{d}x \)
    \(\displaystyle \int \sin(x)\sin\bigl(\cos(x)\bigr) \,\mathrm{d}x \)
    \(\displaystyle \int \frac{\cos(3x)}{\sin(3x)} \,\mathrm{d}x \)
    \(\displaystyle \int \mathrm{e}^x\sin(x) \,\mathrm{d}x \)
    \(\displaystyle \int \mathrm{e}^{5x}\sin(7x) \,\mathrm{d}x \)
    \(\displaystyle \int \mathrm{e}^x\cos\bigl(\mathrm{e}^x\bigr) \,\mathrm{d}x \)
    \(\displaystyle \int \mathrm{e}^x\sin\bigl(2^x\bigr) \,\mathrm{d}x \)
    \(\displaystyle \int \mathrm{e}^{2x}\cos\bigl(\mathrm{e}^{-x}\bigr) \,\mathrm{d}x \)
    \(\displaystyle \int \ln\bigl(\sin(x)\bigr)\cos(x) \,\mathrm{d}x \)
    \(\displaystyle \int \cos(x)\mathrm{e}^{\sin(x)}\,\mathrm{d}x \)
    \(\displaystyle \int x^5\cos\bigl(x^3\bigr) \,\mathrm{d}x \)
    \(\displaystyle \int \frac{\cos\bigl(\tfrac{2}{x}\bigr)}{x}\,\mathrm{d}x \)
    \(\displaystyle \int \cos(x)\cos\bigl(\sin(x)\bigr)\sin\Bigl(\sin\bigl(\sin(x)\bigr)\Bigr)\,\mathrm{d}x \)