Integrals of Trigonometric Functions — Worksheet

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Formulae

  1. $\displaystyle \int \sin x\,dx= -\cos x + C$
  2. $\displaystyle \int \cos x\,dx= \sin x + C$
  3. $\displaystyle \int \tan x\,dx= \ln|\sec x| + C$
  4. $\displaystyle \int \cot x\,dx= \ln|\sin x| + C$
  5. $\displaystyle \int \sec x\,dx= \ln|\sec x + \tan x| + C$
  6. $\displaystyle \int \csc x\,dx= \ln|\csc x - \cot x| + C = -\,\ln|\csc x + \cot x| + C$
  7. $\displaystyle \int \sec^2 x\,dx= \tan x + C$
  8. $\displaystyle \int \csc^2 x\,dx= -\cot x + C$
  9. $\displaystyle \int (\sec x\tan x)\,dx= \sec x + C$
  10. $\displaystyle \int (\csc x\cot x)\,dx= -\csc x + C$
Angle change rule. If the angle is $(ax+b)$, then divide by $a$.
Example: $\displaystyle \int \sin(ax+b)\,dx= -\tfrac{1}{a}\cos(ax+b)+C$.

Bonus identities

  • $\displaystyle \cos 4x = 8\cos^4 x - 8\cos^2 x + 1$
  • $\displaystyle \cos 4x = 8\sin^4 x - 8\sin^2 x + 1$

Practice A — Evaluate the integrals. Include + C.

  1. $\displaystyle \int \sin\!\left(3x-\tfrac{\pi}{6}\right)\,dx$
  2. $\displaystyle \int \cos(5x)\,dx$
  3. $\displaystyle \int \tan x \, dx$
  4. $\displaystyle \int \cot x \, dx$
  5. $\displaystyle \int \sec x \, dx$
  6. $\displaystyle \int \csc x \, dx$
  7. $\displaystyle \int \sec^2(2x+1)\,dx$
  8. $\displaystyle \int \csc^2(4x)\,dx$
  9. $\displaystyle \int (\sec x\tan x)\,dx$
  10. $\displaystyle \int (\csc x\cot x)\,dx$

Practice B — Verify by differentiation.

  1. Show that $\displaystyle \frac{d}{dx}\left[-\tfrac{1}{7}\cos(7x)\right]=\sin(7x)$.
  2. Show that $\displaystyle \frac{d}{dx}\big[\ln|\sec x+\tan x|\big]=\sec x$.