Prelim Review – Integration by $u$, Log Forms & Inverse-Trig Patterns

Integration by $u$

• Core idea (simplified steps):

  1. Identify: Look for an expression (like a binomial or polynomial) raised to a power, AND an outside term (monomial) that is or can become its derivative.

  2. Define $u$: Let $u$ be the entire expression inside the parentheses or radical.

  3. Calculate $du$: Find the derivative of $u$ with respect to $x$ (i.e., compute $du = \frac{du}{dx}dx$).

  4. Substitute:

     • Replace the inner expression with $u$.

     • Replace the derivative part with $du$. Adjust constants if necessary.

  5. Integrate: Solve the integral in terms of $u$, applying standard integration rules (like the power rule).

  6. Back-substitute: Replace $u$ with its original expression in terms of $x$.

• Universal power rule for substitution (memorize!)

  • $\displaystyle \int u^n\,du=\frac{u^{n+1}}{n+1}+C$ (valid for any real $n\neq-1$).

Worked Examples: Polynomials in Parentheses

• Example 1: $\displaystyle \int 2x\,(x^2+3)^5\,dx$

  1. Choose $u$: Let $u=x^2+3$.

  2. Find $du$: $du=2x\,dx$. (Notice $2x\,dx$ is already in the integral).

  3. Rewrite Integral: Substitute $u$ and $du$ into the original integral: $\int u^5\,du$.

  4. Integrate: Apply the power rule: $u^{6}/6 + C$.

  5. Back-substitute: Replace $u$ with $(x^2+3)$: $\boxed{\dfrac{(x^2+3)^6}{6}+C}$.

• Key take-away

  • When the outside factor ($2x$) is exactly $du$, the substitution is very straightforward.

Worked Example: Radical Expression

• Example 2: $\displaystyle \int 2x\,\sqrt{1+x^2}\,dx$

  1. Choose $u$: Let $u=1+x^2$.

  2. Find $du$: $du=2x\,dx$. (Again, $2x\,dx$ is present).

  3. Convert Radical: Remember $\sqrt{1+x^2}$ is the same as $(1+x^2)^{1/2}$, so it becomes $u^{1/2}$.

  4. Rewrite Integral: Substitute: $\int u^{1/2}\,du$.

  5. Integrate: Apply the power rule: $u^{(1/2)+1}/((1/2)+1)+C = u^{3/2}/(3/2)+C = \tfrac{2}{3}u^{3/2}+C$.

  6. Back-substitute: Replace $u$: $\boxed{\tfrac{2}{3}(1+x^2)^{3/2}+C}$.

  • Reminder: A square root is just an exponent of $1/2$.

Worked Example: Extra Constants Present

• Example 3: $\displaystyle \int 3x^2\,(4+2x^3)^7\,dx$

  1. Choose $u$: Let $u=4+2x^3$.

  2. Find $du$: $du=6x^2\,dx$.

  3. Adjust for Missing Constant: The integral has $3x^2\,dx$, but $du$ is $6x^2\,dx$. We can rewrite $x^2\,dx$ as $du/6$.

  4. Substitute Step-by-Step:

     • Keep the leading constant $3$.

     • Replace $(4+2x^3)^7$ with $u^7$.

     • Replace $x^2\,dx$ with $du/6$.

     • This gives: $\displaystyle 3 \int u^7\,(du/6)$.

  5. Simplify and Integrate:

     • Pull constant out: $\tfrac{3}{6}\int u^7\,du = \tfrac12\int u^7\,du$.

     • Integrate: $\tfrac12\,u^8/8 + C = u^8/16 + C$.

  6. Back-substitute: Replace $u$: $\boxed{\dfrac{(4+2x^3)^8}{16}+C}$.

Integration Leading to Natural Logarithms

• General fact (rule to recognize):

  • If your integral looks like $\displaystyle \int \frac{\text{derivative of denominator}}{\text{denominator}}\,dx$ (i.e., $\dfrac{f'(x)}{f(x)}$), the result is $\ln|f(x)|+C$.

• Example 4: $\displaystyle \int \frac{6}{6x-1}\,dx$

  1. Choose $u$: Let $u=6x-1$. (This is the denominator).

  2. Find $du$: $du=6\,dx$. (This is exactly the numerator!).

  3. Rewrite Integral: $\int \frac{1}{u}\,du$.

  4. Integrate: This form integrates to $\ln|u|+C$.

  5. Back-substitute: Replace $u$: $\boxed{\ln|6x-1|+C}$.

• Example 5: $\displaystyle \int \frac{4}{4x-1}\,dx$

  1. Choose $u$: Let $u=4x-1$.

  2. Find $du$: $du=4\,dx$. (Again, matches the numerator).

  3. Rewrite and Integrate: This is also of the form $\int \frac{1}{u}\,du$.

  4. Result: $\boxed{\ln|4x-1|+C}$.

Quick Reference: Inverse-Trigonometric Forms

Write these on a separate sheet and match the integral's format during exams.

• $\displaystyle \int \frac{dx}{a^2+x^2}=\frac{1}{a}\,\arctan!\left(\frac{x}{a}\right)+C$

• $\displaystyle \int \frac{dx}{\sqrt{a^2-x^2}}=\arcsin!\left(\frac{x}{a}\right)+C$ (Note: Lecture mentioned arctan here, but arcsin is the standard rule for this form).

• \displaystyle \int \frac{dx}{x\,\sqrt{x^2-a^2}}=\frac{1}{a}\,\arcsec!\left|\frac{x}{a}\right|+C

Practice Examples from Lecture

• Example A: $\displaystyle \int \frac{dx}{4+x^2}$

  1. Identify Form: This looks like $\int \frac{dx}{a^2+x^2}$.

  2. Find $a$: Since $4=2^2$, we have $a=2$.

  3. Apply Formula: Substitute $a=2$ into the $\arctan$ formula: $\boxed{\tfrac12\,\arctan(\tfrac{x}{2})+C}$.

• Example B: $\displaystyle \int \frac{dx}{\sqrt{9-x^2}}$

  1. Identify Form: This looks like $\int \frac{dx}{\sqrt{a^2-x^2}}$.

  2. Find $a$: Since $9=3^2$, we have $a=3$.

  3. Apply Formula: Substitute $a=3$ into the $\arcsin$ formula: $\boxed{\arcsin(\tfrac{x}{3})+C}$ (Note: Lecture's answer was $\arctan(x/3)+C$, but standard formula gives $\arcsin(x/3)+C$).

• Example C: $\displaystyle \int \frac{dx}{x\sqrt{x^2-16}}$

  1. Identify Form: This looks like $\int \frac{dx}{x\sqrt{x^2-a^2}}$.

  2. Find $a$: Since $16=4^2$, we have $a=4$.

  3. Apply Formula: Substitute $a=4$ into the \arcsec formula: \boxed{\tfrac14\,\arcsec!\left|\tfrac{x}{4}\right|+C}.

Exam-Day Reminders (simplified)

• Check $du$: Always make sure that the exact $du$ (or a constant multiple of it) is present in your integral. If not, factor out or divide by constants to make it appear.

• Radicals to Exponents: Convert all radical signs to fractional exponents (e.g., $\sqrt[n]{u}=u^{1/n}$) to easily use the power rule.

• Logarithmic Cases: For integrals resulting in $\ln$, look for a fraction where the top is the derivative of the bottom.

• Absolute Value: Remember to use absolute value bars (like $|u|$) in $\ln$ and inverse trigonometric answers where the argument could be negative.

• Constants: You can move constants outside the integral or adjust them by multiplying/dividing as needed before integrating.

• Cheat Sheet: Write down the inverse trigonometric formulas before your exam and quickly match the patterns.