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

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

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

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

Flashcard 4: What is $du$ if $u = e^{x^2}$?

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

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

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

Flashcard 6: Identify $du$ for substitution when $u = e^{x} + x$.

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

Flashcard 7: 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 8: 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 9: 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 10: What is $du$ if $u = x^2 - 2x + 1$?

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

Flashcard 11: What is $du$ if $u = e^{x^2}$?

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

Flashcard 12: Identify $du$ for substitution when $u = e^{x} + x$.

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

Flashcard 13: What is $du$ if $u = x^3 + 5x$?

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

Flashcard 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: 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.