Integrating Using Substitution - AP Calculus BC

1. Identify the most effective choice of $u$ in $\int (5x-1)^7\,dx$.

   $u=5x-1$. The derivative of $5x-1$ is $5$, a constant factor.

2. Identify $du$ for substitution when $u = \frac{1}{x}$.

   $du = -\frac{1}{x^2} dx$. Differentiate $u = \frac{1}{x}$ using power rule.

3. Evaluate $\int \cos(3x)\,dx$ using substitution.

   $\frac{1}{3}\sin(3x)+C$. Since $du=3\,dx$, we get $\frac{1}{3}\int\cos(u)\,du$.

4. What is the substitution rule for definite integrals, including how to change limits?

   $\int_a^b f(g(x))g'(x)\,dx=\int_{u(a)}^{u(b)} f(u)\,du$. Change limits: when $x=a$, $u=g(a)$; when $x=b$, $u=g(b)$.

5. Evaluate $\int_0^1 2x\,e^{x^2}\,dx$ by substitution with changed limits.

   $e-1$. With $u=x^2$, limits change from $[0,1]$ to $[0,1]$; integral is $e^u|_0^1$.

6. Evaluate $\int \frac{e^{2x}}{1+e^{2x}}\,dx$ using substitution.

   $\frac{1}{2}\ln\left|1+e^{2x}\right|+C$. Becomes $\frac{1}{2}\int\frac{du}{u}$ after substitution.

7. Identify the most effective choice of $u$ in $\int \sec^2(4x)\,dx$.

   $u=4x$. The derivative of $4x$ is $4$, a constant we can factor out.

8. What is $du$ if $u = 3x^2 - 4$?

   $du = 6x \, dx$. Differentiate $u = 3x^2 - 4$ to get $du$.

9. Identify the most effective choice of $u$ in $\int \frac{2x}{x^2+5}\,dx$.

   $u=x^2+5$. The derivative of $x^2+5$ is $2x$, which appears in the numerator.

10. What is $du$ if $u = \tan^{-1}(x^2)$?

    $du = \frac{2x}{1+x^4} \, dx$. Differentiate $u = \tan^{-1}(x^2)$ using chain rule.

11. Identify $du$ for substitution when $u = e^{x} + x$.

    $du = (e^{x} + 1) \, dx$. Differentiate $u = e^x + x$ to get $du$.

12. Identify $du$ for substitution when $u = \tan^{-1}(e^{x})$.

    $du = \frac{e^{x}}{1+e^{2x}} \, dx$. Differentiate $u = \tan^{-1}(e^x)$ using chain rule.

13. Identify the most effective choice of $u$ in $\int x\sqrt{x^2+9}\,dx$.

    $u=x^2+9$. The derivative of $x^2+9$ is $2x$, and we have $x$ as a factor.

14. Evaluate $\int \frac{x}{\sqrt{x^2+4}}\,dx$ using substitution.

    $\sqrt{x^2+4}+C$. Becomes $\frac{1}{2}\int u^{-1/2}\,du = u^{1/2}+C$.

15. Evaluate $\int \sec^2(4x)\,dx$ using substitution.

    $\frac{1}{4}\tan(4x)+C$. Since $\int\sec^2(u)\,du=\tan(u)+C$ and $du=4\,dx$.

16. Evaluate $\int x\sqrt{x^2+9}\,dx$ using substitution.

    $\frac{1}{3}(x^2+9)^{\frac{3}{2}}+C$. Becomes $\frac{1}{2}\int u^{1/2}\,du$ using power rule.

17. What is $du$ if $u = x^3 + 5x$?

    $du = (3x^2 + 5)dx$. Differentiate $u = x^3 + 5x$ to get $du$.

18. What is $du$ if $u = x^2 - 2x + 1$?

    $du = (2x - 2) \, dx$. Differentiate $u = x^2 - 2x + 1$ to get $du$.

19. Evaluate $\int \frac{2x}{x^2+5}\,dx$ using substitution.

    $\ln\left|x^2+5\right|+C$. Since $du=2x\,dx$, this becomes $\int\frac{du}{u}=\ln|u|+C$.

20. Find and correct the missing factor error: $\int (x^2+1)^5\,dx=\frac{(x^2+1)^6}{6}+C$.

    Correct: $\int x(x^2+1)^5\,dx=\frac{(x^2+1)^6}{12}+C$. The integrand has factor $x$, which is half of $(x^2+1)'=2x$.

21. Identify the most effective choice of $u$ in $\int \frac{x}{\sqrt{x^2+4}}\,dx$.

    $u=x^2+4$. The derivative of $x^2+4$ is $2x$, and we have $x$ in the numerator.

22. Evaluate $\int (5x-1)^7\,dx$ using substitution.

    $\frac{(5x-1)^8}{40}+C$. Using power rule: $\frac{1}{5}\cdot\frac{u^8}{8}$ with $u=5x-1$.

23. What is $du$ if $u = e^{x^2}$?

    $du = 2xe^{x^2} \, dx$. Differentiate $u = e^{x^2}$ using chain rule.

24. What is the substitution rule for indefinite integrals (the $u$-substitution formula)?

    $\int f(g(x))g'(x)\,dx=\int f(u)\,du$ with $u=g(x)$. Substituting $u=g(x)$ and $du=g'(x)dx$ transforms the integral.

25. Identify $du$ for substitution when $u = x^4 + x^2$.

    $du = (4x^3 + 2x) \, dx$. Differentiate $u = x^4 + x^2$ to get $du$.

26. Identify the most effective choice of $u$ in $\int \cos(3x)\,dx$.

    $u=3x$. The derivative of $3x$ is $3$, a constant factor we can adjust for.

27. Identify the most effective choice of $u$ in $\int \frac{e^{2x}}{1+e^{2x}}\,dx$.

    $u=1+e^{2x}$. The derivative of $1+e^{2x}$ is $2e^{2x}$, nearly matching the numerator.

28. What is the fastest test to confirm an integrand matches $f(g(x))g'(x)$ for substitution?

    Check for an inner function $g(x)$ and its derivative $g'(x)$ as a factor. Look for $f(g(x))$ structure where $g'(x)$ appears as a factor.

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