AP Calculus BC Flashcards: Integrating Using Substitution

What this deck covers

This deck focuses on Integrating Using Substitution, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: Identify the most effective choice of $u$ in $\int (5x-1)^7\,dx$.

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

Flashcard 3: Identify $du$ for substitution when $u = \frac{1}{x}$.

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

Flashcard 4: Identify $du$ for substitution when $u = e^{x} + x$.

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

Flashcard 5: Evaluate $\int \cos(3x)\,dx$ using substitution.

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

Flashcard 6: What is $du$ if $u = x^3 + 5x$?

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

Flashcard 7: What is the substitution rule for definite integrals, including how to change limits?

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

Flashcard 8: Evaluate $\int_0^1 2x\,e^{x^2}\,dx$ by substitution with changed limits.

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

Flashcard 9: Evaluate $\int \frac{e^{2x}}{1+e^{2x}}\,dx$ using substitution.

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

Flashcard 10: Identify the most effective choice of $u$ in $\int \sec^2(4x)\,dx$.

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

Flashcard 11: Identify $du$ for substitution when $u = \frac{1}{x}$.

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

Flashcard 12: What is $du$ if $u = 3x^2 - 4$?

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

Flashcard 13: Identify the most effective choice of $u$ in $\int \frac{2x}{x^2+5}\,dx$.

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

Flashcard 14: What is $du$ if $u = \tan^{-1}(x^2)$?

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

Flashcard 15: Identify $du$ for substitution when $u = x^4 + x^2$.

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

Flashcard 16: Identify $du$ for substitution when $u = e^{x} + x$.

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

Flashcard 17: What is $du$ if $u = e^{x^2}$?

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

Flashcard 18: Identify $du$ for substitution when $u = \tan^{-1}(e^{x})$.

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

Flashcard 19: Identify the most effective choice of $u$ in $\int x\sqrt{x^2+9}\,dx$.

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

Flashcard 20: Evaluate $\int \frac{x}{\sqrt{x^2+4}}\,dx$ using substitution.

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

Flashcard 21: Evaluate $\int \sec^2(4x)\,dx$ using substitution.

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

Flashcard 22: Evaluate $\int x\sqrt{x^2+9}\,dx$ using substitution.

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

Flashcard 23: What is $du$ if $u = x^3 + 5x$?

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

Flashcard 24: What is $du$ if $u = x^2 - 2x + 1$?

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

Flashcard 25: Evaluate $\int \frac{2x}{x^2+5}\,dx$ using substitution.

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

Flashcard 26: What is $du$ if $u = x^2 - 2x + 1$?

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

Flashcard 27: Find and correct the missing factor error: $\int (x^2+1)^5\,dx=\frac{(x^2+1)^6}{6}+C$.

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

Flashcard 28: Identify the most effective choice of $u$ in $\int \frac{x}{\sqrt{x^2+4}}\,dx$.

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

Flashcard 29: Evaluate $\int (5x-1)^7\,dx$ using substitution.

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

Flashcard 30: What is $du$ if $u = e^{x^2}$?

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

Flashcard 31: Identify $du$ for substitution when $u = \tan^{-1}(e^{x})$.

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

Flashcard 32: What is the substitution rule for indefinite integrals (the $u$-substitution formula)?

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

Flashcard 33: Identify $du$ for substitution when $u = x^4 + x^2$.

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

Flashcard 34: Identify the most effective choice of $u$ in $\int \cos(3x)\,dx$.

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

Flashcard 35: Identify the most effective choice of $u$ in $\int \frac{e^{2x}}{1+e^{2x}}\,dx$.

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

Flashcard 36: What is the fastest test to confirm an integrand matches $f(g(x))g'(x)$ for substitution?

Answer: 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.