6.11 Integrating Using Integration by Parts

    Cards (47)

    • Integration by Parts is a technique used to integrate products of functions
    • Steps to apply Integration by Parts
      1️⃣ Choose uu and dvdv
      2️⃣ Find dudu and vv
      3️⃣ Apply the Formula
      4️⃣ Evaluate the new integral
    • The Integration by Parts formula transforms complex integrals into simpler ones.
    • What is the Integration by Parts formula?
      udv=\int u \, dv =uvvdu uv - \int v \, du
    • In the Integration by Parts formula, du</latex> is the derivative of uu with respect to x
    • Why is choosing uu and dvdv correctly important in Integration by Parts?

      To simplify the new integral
    • The new integral in Integration by Parts is always more complex than the original.
      False
    • The final step in Integration by Parts is to evaluate the new integral
    • What is the formula for Integration by Parts?
      udv=\int u \, dv =uvvdu uv - \int v \, du
    • Integration by Parts is based on the product rule of differentiation.
    • In Integration by Parts, dvdv is typically chosen so that it is easily integrable
    • What is the Integration by Parts formula?
      udv=\int u \, dv =uvvdu uv - \int v \, du
    • What is the Integration by Parts formula?
      udv=\int u \, dv =uvvdu uv - \int v \, du
    • By choosing uu and dvdv appropriately, the new integral vdu\int v \, du is often easier to solve than the original
    • Steps to solve an integral using Integration by Parts
      1️⃣ Choose uu and dvdv
      2️⃣ Find dudu and vv
      3️⃣ Apply the formula
      4️⃣ Evaluate the new integral
    • dv in the Integration by Parts formula should be easily integrable.
    • To find dudu, you need to differentiate uu, and to find vv, you need to integrate <dv</latex><dv</latex>
    • What is the result of choosing u=u =x x and dv=dv =cos(x)dx \cos(x) \, dx in xcos(x)dx\int x \cos(x) \, dx?

      xcos(x)+- x\cos(x) +sin(x)+ \sin(x) +C C
    • Choosing uu as the function that simplifies upon differentiation is a good strategy.
    • Match the function with its derivative or integral:
      u = x ↔️ du = dx
      dv = e^{x} dx ↔️ v = e^{x}
    • What is the value of exdx\int e^{x} dx?

      e^{x} + C</latex>
    • Integration by Parts is used to simplify integrals of products of functions.
    • Steps to solve an integral using Integration by Parts
      1️⃣ Choose uu and dvdv
      2️⃣ Find dudu and vv
      3️⃣ Apply the formula
      4️⃣ Evaluate the new integral
    • What is the Integration by Parts formula?
      udv=\int u \, dv =uvvdu uv - \int v \, du
    • The new integral vdu\int v \, du in Integration by Parts is often easier to solve than the original.
    • What is the Integration by Parts formula?
      udv=\int u \, dv =uvvdu uv - \int v \, du
    • In the Integration by Parts formula, uu and vv are functions of x
    • The term dudu in the Integration by Parts formula represents the derivative of uu with respect to xx.
    • What does dvdv represent in the Integration by Parts formula?

      The differential of vv
    • Steps for applying Integration by Parts
      1️⃣ Choose uu and dvdv
      2️⃣ Find dudu and vv
      3️⃣ Apply the formula
      4️⃣ Evaluate the new integral
    • For the integral xexdx\int x e^{x} dx, we choose u=u =x x and dv = e^{x} dx</latex>, which simplifies to dx
    • What is the derivative of u=u =x x in the example xexdx\int x e^{x} dx?

      du=du =dx dx
    • What is the integral of dv = e^{x} dx</latex> in the example xexdx\int x e^{x} dx?

      v=v =ex e^{x}
    • The final result of xexdx\int x e^{x} dx is xexex+x e^{x} - e^{x} +C C.
    • What does the acronym LIATE stand for in the context of Integration by Parts?
      Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential
    • When using LIATE, the function that appears earlier in the list is chosen as u
    • The acronym LIATE helps in choosing dvdv as the function that is easier to differentiate.

      False
    • In the integral xcos(x)dx\int x \cos(x) \, dx, which function should be chosen as uu?

      u=u =x x
    • How is dudu found in the Integration by Parts process?

      By differentiating uu
    • When finding vv, no constant of integration is needed because it cancels out in the final formula