Integration Techniques and U-Substitution

Power Rule in Integration

  • The power rule states that for a function $u$, the integral is given by:
    ∫un du=un+1n+1+C\int u^n \, du = \frac{u^{n+1}}{n+1} + C
    where $n \neq -1$.
  • To use this rule properly, identify the function $g(x)$ that applies to the given integral.

Using U-Substitution

  • It can be difficult to find $g(x)$ in polynomial integrals.
  • Example of u-substitution:
    • Let $u = \cos(x)$.
    • Then, the derivative $\frac{du}{dx} = -\sin(x)$
  • The negative brings about:
    −du=sin⁡(x) dx-du = \sin(x) \, dx

Integration Application

  • If you encounter $\int \sin(g(x)) g'(x) \, dx$, it simplifies using:
    ∫sin⁡(u) du=−cos⁡(u)+C\int \sin(u) \, du = -\cos(u) + C
  • Thus,
    ∫sin⁡(g(x))g′(x) dx=−cos⁡(g(x))+C\int \sin(g(x)) g'(x) \, dx = -\cos(g(x)) + C

Evaluating More Complex Integrals

  • Example integral: ∫xx−1 dx\int \frac{x}{\sqrt{x-1}} \, dx

  • Let $u = x - 1$. Therefore, $du = dx$ and $x = u + 1$.

  • The integral now transforms to:
    ∫u+1u du=∫(u1/2+u−1/2) du\int \frac{u+1}{\sqrt{u}} \, du = \int (u^{1/2} + u^{-1/2}) \, du

  • Computing this leads to:
    23u3/2+2ln⁡∣u∣+C\frac{2}{3} u^{3/2} + 2 \ln|u| + C
    Replace $u$ back to get the final result.

Important Exponential Functions

  • For $e^{g(x)}$, the derivative is:
    ddx(eg(x))=eg(x)g′(x)\frac{d}{dx}(e^{g(x)}) = e^{g(x)} g'(x)
  • This means:
    ∫eg(x)g′(x) dx=eg(x)+C\int e^{g(x)} g'(x) \, dx = e^{g(x)} + C
  • Similarly, for the base $a$:
    ∫ag(x)g′(x) dx=ag(x)ln⁡(a)+C\int a^{g(x)} g'(x) \, dx = \frac{a^{g(x)}}{\ln(a)} + C

Summary of Integration Techniques

  • Always remember to check if the integral matches any of the standard forms.
  • U-substitution is useful in many cases to simplify functions.
  • Power rule and logarithmic properties can simplify integrals involving exponentials and polynomials.

Evaluating Limits in Integrals

  • Example for definite integrals:
    • ∫12e1−x dx\int_{1}^2 e^{1-x} \, dx
    • Use $u = 1 - x$ ⇒ $dx = -du$. Update limits for $u$.
    • Evaluate the integral with the new limits for $u$.

Final Tips

  • Revise derivatives closely, as they inform integration.
  • Practice different types of integrals to strengthen understanding.
  • Contact the teacher for clarifications and additional exercises before the final exam.