Calculus 2 2026-05-27 - Review: Integration Techniques and Improper Integrals

Advanced Integration Techniques: Integration by Parts and Trigonometric Methods

• Integration by Parts ($u-dv$ Substitution):

  • Formula: $\int u \, dv = uv - \int v \, du$.

  • Example 1: $\int (2x+2)\sin(x) \, dx$

    • Let $u = 2x+2 \Rightarrow du = 2 \, dx$.

    • Let $dv = \sin(x) \, dx \Rightarrow v = -\cos(x)$.

    • Result: $-(2x+2)\cos(x) - \int -2\cos(x) \, dx = -(2x+2)\cos(x) + 2\sin(x) + C$.

  • Example 2: $\int 7t^2 \ln(t) \, dt$ on bounds $[1, e]$.

    • Let $u = \ln(t) \Rightarrow du = \frac{1}{t} \, dt$.

    • Let $dv = 7t^2 \, dt \Rightarrow v = \frac{7}{3}t^3$.

    • Evaluation: $[\ln(t) \times \frac{7}{3}t^3]_1^e - \int_1^e \frac{7}{3}t^3 \times \frac{1}{t} \, dt$.

    • Simplifies to: $\frac{7}{3}e^3 - [\frac{7}{9}t^3]_1^e = \frac{7}{3}e^3 - (\frac{7}{9}e^3 - \frac{7}{9})$.

    • Final Value: $\frac{14}{9}e^3 + \frac{7}{9}$.

• Powers of Sine and Cosine:

  • If one power is odd: Use $u$-substitution. Example: $\int \cos^2(x)\sin(x) \, dx$, let $u = \cos(x)$.

  • If all powers are even: Use double-angle identities.

    • Identity: $\sin^2(nx) = \frac{1}{2}(1 - \cos(2nx))$.

    • Identity: $\cos^2(nx) = \frac{1}{2}(1 + \cos(2nx))$.

  • Example: $\int \sin^2(3x) \, dx$ becomes $\int \frac{1}{2}(1 - \cos(6x)) \, dx$.

    • Result: $\frac{1}{2}x - \frac{1}{12}\sin(6x) + C$.

• Trigonometric Substitution:

  • Used when encountering forms like $\sqrt{a^2 - x^2}$.

  • Let $x = a\sin(\theta)$, then $dx = a\cos(\theta) \, d\theta$.

  • Example: $\int \frac{30x^2}{\sqrt{81 - x^2}} \, dx$

    • Let $x = 9\sin(\theta) \Rightarrow dx = 9\cos(\theta) \, d\theta$.

    • Denominator: $\sqrt{81 - 81\sin^2(\theta)} = 9\cos(\theta)$.

    • The $9\cos(\theta)$ in the denominator cancels with the $dx$ term.

    • Integral becomes: $\int 30(81\sin^2(\theta)) \, d\theta = 2430 \int \sin^2(\theta) \, d\theta$.

    • Apply $\sin^2(\theta) = \frac{1}{2} - \frac{1}{2}\cos(2\theta)$.

    • Back-substitute using the triangle: $\sin(\theta) = \frac{x}{9}$, $\cos(\theta) = \frac{\sqrt{81-x^2}}{9}$.

Partial Fraction Decomposition

• General Logic: Transform complex rational functions into sums of simpler fractions.

• Linear Factors Example: $\frac{4x+44}{x(x+8)} = \frac{A}{x} + \frac{B}{x+8}$.

  • Multiply by denominator: $4x+44 = A(x+8) + Bx$.

  • To solve for $A$, set $x = 0$: $44 = 8A \Rightarrow A = 5.5 \text{ (or } \frac{11}{2})$.

  • To solve for $B$, set $x = -8$: $4(-8)+44 = B(-8) \Rightarrow 12 = -8B \Rightarrow B = -1.5$.

• Matching Coefficients Method: Distribute the constants and group terms by power of $x$ (e.g., all $x^2$ terms together, all constants together) then create a system of equations.

Improper Integrals

• Types of Impropriety:

  • Infinite bounds (e.g., $\int_a^\infty f(x) \, dx$).

  • Discontinuity within the bounds (e.g., vertical asymptote at $x=c$ where a < c < b).

• Handling Discontinuities:

  • Problem: $\int_1^6 \frac{9}{\sqrt[7]{x-2}^8} \, dx$.

  • The function is improper at $x=2$ because the denominator becomes zero.

  • Split the integral: $\int_1^2 \frac{9}{(x-2)^{8/7}} \, dx + \int_2^6 \frac{9}{(x-2)^{8/7}} \, dx$.

  • Use limits: $\lim_{t \to 2^-} \int_1^t \dots + \lim_{t \to 2^+} \int_t^6 \dots$

  • This specific integral diverges because the exponent in the denominator is greater than or equal to 1.

• Infinite Bounds Examples:

  • Example 1: $\int_2^\infty \frac{1}{(3x+1)^{5/4}} \, dx$

    • Rewrite as limit: $\lim_{t \to \infty} \int_2^t (3x+1)^{-5/4} \, dx$.

    • Anti-derivative: $\frac{1}{3} \times (-4)(3x+1)^{-1/4}$.

    • As $t \to \infty$, the term $\frac{1}{(3t+1)^{1/4}}$ approaches $0$.

    • Result: $\frac{4}{3(7)^{1/4}}$ (convergent).

  • Example 2: $\int_{-\infty}^0 \frac{1}{x^2+1} \, dx$

    • Anti-derivative: $\arctan(x)$.

    • Evaluation: $\arctan(0) - \lim_{t \to -\infty} \arctan(t)$.

    • Know the horizontal asymptotes of $\arctan(x)$: as $x \to \infty$, $y = \frac{\pi}{2}$; as $x \to -\infty$, $y = -\frac{\pi}{2}$.

    • Result: $0 - (-\frac{\pi}{2}) = \frac{\pi}{2}$.

  • Example 3: $\int_0^\infty e^{-5x} \, dx$

    • Anti-derivative: $-\frac{1}{5}e^{-5x}$.

    • Evaluation: $\lim_{t \to \infty} [-\frac{1}{5}e^{-5t}] - [-\frac{1}{5}e^0]$.

    • Since $e^{-5t} \to 0$ as $t \to \infty$, the result is $0 - (-\frac{1}{5}) = \frac{1}{5}$.

Questions & Discussion

• Q: How does the Engineering Lab grade work?

  • A: It is a 1-credit hour class, but the workload is intentionally high to filter out students who aren't dedicated. It involves significant team projects like the egg drop.

• Q: What is the limit of $\arctan(x)$ at negative infinity?

  • A: It is $-\frac{\pi}{2}$. This is determined by the vertical asymptotes of the original $\tan(x)$ function ($x = \pm \frac{\pi}{2}$), which become horizontal asymptotes for the inverse function.

• Q: How do we handle $\int e^x \cos(x) dx$?

  • A: Use integration by parts twice. Because $e^x$ stays $e^x$ and $\cos(x)$ cycles back from $\sin(x)$, you eventually get the original integral back on the right side and solve for it algebraically (the "add-back" method).", "title": "Calculus 2 Review: Integration Techniques and Improper Integrals" }