U Substitution

5.5

• the substitution rule for indefinite integrals

  • this is just u sub

    • 
      • The reverse chain rule

  • ex.

    • a) $\int x^2(x^3+5{)}^{9}dx$

    • => let u = $x^{3+5}\therefore du=3x^{2}dx$
      • => $\int \frac{1}{3}{u}^{9}du$
        • => $\frac{1}{3}\frac{u^{10}}{10}+c⟹\frac{1}{3}\frac{{x^3+5}^{10}}{10}+c$

    • b) $\int \frac{1}{\sqrt{3-5t}}dt$

      • => let u = $u=3-5t⟹du=-5dt$
        • =>$\int -\frac{1}{5}\frac{1}{\sqrt{u}}du$
          • => $-\frac{2}{5}{u}^{1/2}+c$
            • => $-\frac{2}{5}\sqrt{3-5t}+c$

    • c) $\int \sin (3t)dt$

      • => $u=3t⟹du=3dt$
        • =>$\frac{1}{3}\int \sin (u)du$

    • d) $\int \frac{z^2}{\sqrt{1-z}}dz$

      • $⟹u=1-z⟹du=-dz⟹z=izu$
        • $-\int \frac{z^2}{\sqrt{u}}du⟹-\int \frac{{1-u}^2}{\sqrt{u}}du$
          • =>$-\int u^{-1/2}\ast (1-2u+u^2)\ast du$
            • =>$-2u^{1/2}+\frac{4}{3}{u}^{3/2}-\frac{2}{5}{u}^{5/2}+c$

• The substitution rule for definite integral

  • 

    • Instead of going back to the original variable we can just plug in the output values of “u”

  • Ex.

    • a) ${\int }_{\pi }^{2\pi }\cos 3tdt$

      • $⟹u=3t⟹du=3dt$