patulong po i solve tong mga problems (Integral Calculus)

Jhun Vert's picture

Number 1
$\displaystyle \int \dfrac{(w - 5) \, dw}{w^2 + 5}$

      $\displaystyle = \int \left( \dfrac{w}{w^2 + 5} - \dfrac{5}{w^2 + 5} \right) \, dw$

      $\displaystyle = \frac{1}{2}\int \dfrac{2w \, dw}{w^2 + 5} - 5\int \dfrac{dw}{w^2 + \left( \sqrt{5} \right)^2}$

      $\displaystyle = \frac{1}{2}\int \dfrac{2w \, dw}{w^2 + 5} - 5\int \dfrac{dw}{w^2 + \left( \sqrt{5} \right)^2}$

      $= \dfrac{1}{2}\ln (w^2 + 5) - \dfrac{5}{\sqrt{5}} \arctan \left( \dfrac{w}{\sqrt{5}} \right) + C$

      $= \frac{1}{2}\ln (w^2 + 5) - \sqrt{5} \arctan \left( \dfrac{w}{\sqrt{5}} \right) + C$

sir question po. pano po naging 2wdw yung numerator nung 2nd part?

Teng Kuling's picture

kasi yung derivative ng w^2+5 is 2w.. kaso w lang yung nasa numerator, so dapat lagyan ng 2 at tsaka imu-multiply by 1/2 sa labas ng integral sign para ma neutralize yung 2 sa loob..

elnerjhun24's picture

u=w2+5
du=2wdw
1/2 du=wdw

elnerjhun24's picture

u=w2+5
du=2wdw
1/2 du=wdw