Calculus: Integration by Substitution and the General Power Rule

The General Power Rule for Integration

• Technique Overview: Finding antiderivatives for more complicated functions often requires a technique called integration by substitution, which is based on the Chain Rule for differentiation.

• Conceptual Foundation: When looking for a function whose derivative is $2x(x^2 + 1)^3$, the key observation is the presence of the factor $2x$. Because $2x$ is the exact derivative of $(x^2 + 1)$, the integrand can be viewed in the form $u^n \frac{du}{dx}$.

• Defining the General Power Rule: The rule states that for an integrand containing a factor $u$ raised to a power $n$, the derivative $du/dx$ must also be a factor of the integrand.

• Substitution Notation: Letting $u = x^2 + 1$ and $du = 2x\,dx$, the integral is rewritten:     $\int (x^2 + 1)^3 2x\,dx = \int u^3\,du = \frac{u^4}{4} + C = \frac{(x^2 + 1)^4}{4} + C$

• Role of the Derivative: The technique is effective specifically because the inner function's derivative is present in the integrand. If the derivative factor is missing or different, the General Power Rule cannot be applied directly without modification.

Adjusting the Integrand: Multiplying and Dividing by a Constant

• Missing Constant Adjustment: In many cases, the derivative $du/dx$ is partially missing from the integrand. If the missing part is a constant factor, you can adjust the integrand to fit the General Power Rule.

• Example 2: Integration with Constant Manipulation: Find $\int x(3 - 4x^2)^2 \,dx$.

  • Step 1: Let $u = 3 - 4x^2$.

  • Step 2: Calculate the derivative $du/dx = -8x$. The integrand currently contains $x$, but it needs $-8x$.

  • Step 3: Multiply and divide the integrand by the constant $-8$. The dividing factor $-1/8$ is moved outside the integral sign.

  • Step 4: Integrate using the General Power Rule:         $\int x(3 - 4x^2)^2\,dx = -\frac{1}{8} \int (3 - 4x^2)^2(-8x)\,dx$         $( - \frac{1}{8} ) [ \frac{(3 - 4x^2)^3}{3} ] + C$         $-\frac{(3 - 4x^2)^3}{24} + C$

• Rule for Variables: This strategy of multiplying and dividing only works for constants. You cannot multiply and divide by a variable to create the necessary $du/dx$ factor.

Limitations and Alternative Strategies

• Failure of the General Power Rule (Example 4): Find $\int -8(3 - 4x^2)^2 \,dx$.

  • In this case, letting $u = 3 - 4x^2$ requires $du/dx = -8x$.

  • Although there is a $-8$ in the integrand, the variable $x$ is missing. Because you cannot factor a variable out of an integral sign, you cannot "create" the missing $x$.

  • Solution Strategy: To solve this indefinite integral, expand the integrand: $(3 - 4x^2)^2 = 9 - 24x^2 + 16x^4$. Then distribute the $-8$ and use the Simple Power Rule on each term individually.

• Moving Extra Constant Factors (Example 5): Find $\int 7x^2 \sqrt{x^3 + 1} \,dx$.

  • Step 1: Identify $u = x^3 + 1$. The required derivative is $du/dx = 3x^2$.

  • Step 2: Notice the extra constant factor of $7$. Move this factor outside the integral sign: $7 \int x^2(x^3 + 1)^{1/2}\,dx$.

  • Step 3: Multiply and divide by $3$ to create the required derivative factor.         $\frac{7}{3} \int (x^3 + 1)^{1/2}(3x^2)\,dx$

  • Step 4: Integrate:         $\frac{7}{3} [ \frac{(x^3 + 1)^{3/2}}{3/2} ] + C = \frac{7}{3} [ \frac{2}{3}(x^3 + 1)^{3/2} ] + C = \frac{14}{9}(x^3 + 1)^{3/2} + C$

Integration by Substitution: Change of Variables

• The Procedure: For more complex integrands where the General Power Rule is difficult to recognize, the substitution or "change of variables" procedure is used to completely rewrite the integral in terms of a new variable $u$.

• Relationship to the General Power Rule: If $u = f(x)$, then $du = f'(x)\,dx$. The integral form is $\int g(u)\,du$.

• Guidelines for Integration by Substitution:

  1. Identify u: Let $u$ be a function of $x$ (typically the part of the integrand inside a power, radical, or denominator).

  2. Rewrite Integral: Replace all instances of $x$ and $dx$ with $u$ and $du$.

  3. Integrate: Find the resulting integral in terms of the variable $u$.

  4. Resubstitute: Rewrite the antiderivative as a function of $x$ by replacing $u$ with the original function of $x$.

  5. Check: Differentiate the final result to ensure it matches the original integrand.

• Example 7: Integration by Substitution Implementation: Find $\int x\sqrt{x^2 - 1}\,dx$.

  • Let $u = x^2 - 1$. Then $du = 2x\,dx$, which implies $\frac{1}{2}\,du = x\,dx$.

  • Substitute: $\int \sqrt{u} (\frac{1}{2}) \,du = \frac{1}{2} \int u^{1/2}\,du$.

  • Integrate: $\frac{1}{2} [ \frac{u^{3/2}}{3/2} ] + C = \frac{1}{3}u^{3/2} + C$.

  • Resubstitute: $\frac{1}{3}(x^2 - 1)^{3/2} + C$.

  • Verification: $\frac{d}{dx} [ \frac{1}{3}(x^2 - 1)^{3/2} + C ] = \frac{1}{3}(\frac{3}{2})(x^2 - 1)^{1/2}(2x) = x(x^2 - 1)^{1/2} = x\sqrt{x^2 - 1}$.

Extended Application: Propensity to Consume

• Economic Context (2013 Data): In $2013$, the U.S. poverty level for a family of four was approximately $23,900$. Families at this level tend to consume $100\%$ of their income for necessities (food, clothing, shelter).

• Consumption Behavior: As income increases, the average consumption percentage typically drops below $100\%$.

  • Example: A family earning $30,000$ might save $600$ and consume $29,400$ ($98\%$ of income).

• Marginal Propensity to Consume: This is defined as the rate of change of consumption with respect to income.

• Example 8: Analyzing Consumption: For a family of four in 2013, the marginal propensity to consume income $x$ (in dollars) is modeled by:     $\frac{dQ}{dx} = 0.98(x - 23,899)^{-0.02}$     where $Q$ represents income consumed.

• Finding the Consumption Model:

  • Step 1: Integrate the rate of change:         $Q = \int 0.98(x - 23,899)^{-0.02}\,dx$         Using substitution where $u = x - 23,899$ and $du = dx$:         $Q = 0.98 \int u^{-0.02}\,du = 0.98 [ \frac{u^{0.98}}{0.98} ] + C = (x - 23,899)^{0.98} + C$

  • Step 2: Solve for the integration constant $C$ using the initial condition. At the poverty level, income $x = 23,900$ results in consumption $Q = 23,900$.         $23,900 = (23,900 - 23,899)^{0.98} + C$         $23,900 = 1^{0.98} + C$         $23,900 = 1 + C \implies C = 23,899$

  • Step 3: Final Model: $Q = (x - 23,899)^{0.98} + 23,899$.

• Estimation for Specific Income: For a family with an income of $x = 37,000$:     $Q = (37,000 - 23,899)^{0.98} + 23,899$     $Q \approx 10,838 + 23,899 \approx 34,737$     The family consumes approximately $34,737$ out of their $37,000$ income.

• Visualizing Consumption: Figure 5.4 displays the graph of $Q$, showing the split between "Income consumed" and "Income saved" as income $x$ increases beyond the poverty point intersection at $(23,900, 23,900)$ to the estimated point $(37,000, 34,737)$.