Trigonometric Integrals

Unit 7.2 Of Math 152

Ex.

• a. $\int {\sin }^{2}3xdx$

  • we can use the double angle identity to simplify the integrand ${\cos }_{}2\theta =1-2{\sin }^2\theta$ => ${\sin }^2\theta =\frac{1-\cos 2\theta }{2}$
    • => $\int \frac{1-\cos 6x}{2}dx$ => $\frac{1}{2}x-\frac{\sin 6x}{12}+C$

• b. $\int {\cot }^{2}3xdx⟹\int \frac{{\cos }^{2}3x}{{\sin }^{2}3x}dx⟹\int \frac{1-{\sin }^{2}3x}{{\sin }^{2}3x}dx$

  • => $\int {\csc }^{2}3xdx-\int 1dx=\frac{-\cot (3x)}{2}-x+C$

Products of sines and cosines

• To evaluate $\int {\sin }^{n}x{\cos }^{m}xdx$, there are only two possibilities

  • (easy case) 1. use ${\sin }^2+{\cos }^2=1$ 2. u-sub (Let u equal the even function)

    • At least one of the numbers n and m is odd

      • $\int {\sin }^{3}x{\cos }^{2}xdx$=
        • $\int \sin x(1-{\cos }^{2}x){\cos }^{2}xdx$
          • => let u = cosx, du = -sinx dx
            • => $-\int (1-u^2)u^{2}du$ => $\int u^4-u^{2}du$ => $\frac{{\cos }^{5}x}{5}-\frac{{\cos }^{3}x}{3}+C$

  • (harder case) 1. use ${\sin }^2+{\cos }^2=1$ 2. use the Reduction Formulas

    • Both n and m are even

      • $\int {\sin }^{2}x{\cos }^{2}xdx$
        • => $\int (1-{\cos }^{2}x){\cos }^{2}xdx⟹\int {\cos }^{2}xdx-\int {\cos }^{4}xdx$
          • => $\int {\cos }^{2}xdx-\left(\frac{1}{4}{\cos }^{3}x\sin x+\frac{3}{4}\int {\cos }^{2}xdx\right)$
            • => $\frac{1}{4}\int {\cos }^{2}xdx-\frac{1}{4}{\cos }^{3}x\sin x$
              • => $\frac{1}{4}\left(\frac{1}{2}\cos x\sin x+\frac{1}{2}x\right)-\frac{1}{4}{\cos }^{3}x\sin x+C$

• Ex.

  • a. $\int {\cos }^{5}xdx$

    • => $\int {\cos }^{5}xdx=\int \cos x(1-{\sin }^{2}x{)}^{2}dx$
      • let u = sinx, du = cosx
        • => $\int (1-u^2{)}^{2}du$
          • =>$\int 1-2u^2+u^{4}du⟹u-\frac{2}{3}{u}^3+\frac{u^5}{5}+C$

• Integrating other trig functions:

  • a. $\int \tan xdx$

    • => $\int \frac{\sin x}{\cos x}dx$
      • => -$\int du=dx$