AP Calculus AB Integration by Substitution Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Integration by Substitution – Maths PDF

Cover of the book AP Calculus AB Math Tests with Answer Keys (PDF)

Download as a PDFThe AP Calculus AB tests book with answer keysPrintable PDFSee the book ›

Sommaire

This u substitution test answers for AP Calculus AB covers integration by substitution (Common Core AP CALC AB 6.9, AP CALC AB 6.10, AP CALC AB 6.11, AP CALC AB 6.12, AP CALC AB 6.14), with a free PDF to print.

Use this answer key to score the u-substitution test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State Standards: AP CALC AB 6.9, AP CALC AB 6.10, AP CALC AB 6.11, AP CALC AB 6.12, AP CALC AB 6.14

U substitution test answers: worked solutions

Question 1 – Indefinite substitution [2 pts]

Let \(u=x^{3}-5\), \(du=3x^{2}dx\). \(\int u^{4}du=\frac{u^{5}}{5}\).

Final answer: \(\frac{(x^{3}-5)^{5}}{5}+C\).

Scoring: 1 point for u and du; 1 point for the final antiderivative.

Question 2 – Definite substitution [3 pts]

Let \(u=x^{2}\), \(du=2x\,dx\). Limits: \(0\to 0\), \(2\to 4\).

\(\int_{0}^{4}e^{u}du=e^{4}-1\approx 53.6\).

Final answer: \(e^{4}-1\).

The graph of f(x) = 2x e^(x²).

Scoring: 1 point for u, du and limits; 1 point for the antiderivative; 1 point for the value.

Question 3 – Exponential substitution [2 pts]

Let \(u=3x+1\), \(dx=\frac{1}{3}du\). The integral is \(\frac{1}{3}e^{u}\).

Final answer: \(\frac{1}{3}e^{3x+1}+C\).

Scoring: 1 point for the substitution; 1 point for the result.

Question 4 – Trigonometric substitution [3 pts]

Let \(u=\sin x\), \(du=\cos x\,dx\). Limits: \(0\to 0\), \(\frac{\pi}{2}\to 1\).

\(\int_{0}^{1}u^{3}du=\frac{1}{4}\).

Final answer: \(\frac{1}{4}\).

Scoring: 1 point for u and du; 1 point for the limits; 1 point for the value.

Question 5 – Logarithm form [3 pts]

Let \(u=1+\ln x\), \(du=\frac{1}{x}dx\). Limits: \(1\to 1\), \(e\to 2\).

\(\int_{1}^{2}\frac{1}{u}du=\ln 2\).

Final answer: \(\ln 2\approx 0.693\).

The graph of f(x) = 1/(x(1 + ln x)).

Scoring: 1 point for u and du; 1 point for the limits; 1 point for the value.

Question 6 – Long division [2 pts]

Divide: \(x^{2}+5=(x+2)(x-2)+9\), so \(\frac{x^{2}+5}{x+2}=x-2+\frac{9}{x+2}\).

Final answer: \(\frac{x^{2}}{2}-2x+9\ln|x+2|+C\).

Scoring: 1 point for the division; 1 point for the antiderivative.

Question 7 – Inflow of water [3 pts]

\(\int_{0}^{5}40te^{-t^{2}/10}\,dt\).
Let \(u=-\frac{t^{2}}{10}\), \(t\,dt=-5\,du\). An antiderivative is \(-200e^{-t^{2}/10}\); \(\left[-200e^{-t^{2}/10}\right]_{0}^{5}=200\left(1-e^{-2.5}\right)\approx 183.6\) gallons.

Inflow rate in gallons per minute.

Scoring: 1 point for the integral; 1 point for the antiderivative; 1 point for the value.

Question 8 – Choosing a technique [2 pts]

u-substitution with \(u=x^{2}\): \(\frac{1}{2}\sin(x^{2})+C\).
Split the fraction (division): \(x+\frac{1}{x}\), so \(\frac{x^{2}}{2}+\ln|x|+C\).

Scoring: 1 point per part.

Total score

Total

20 points

Key facts behind this u substitution test answers

Key facts

Let u = g(x) and du = g′(x) dx, then the integral becomes the integral of f(u) du.
Adjust constant factors: if du is off by a constant, divide or multiply to compensate.
For definite integrals, change the limits to u-values or return to x before evaluating.
Use long division for improper rational functions and completing the square for irreducible quadratics.

How to use this u substitution test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

Download the u substitution test answers as a PDF with the button below and keep it with your notes.

All AP Calculus AB chapters: AP Calculus AB math.

Voter.. post

Free printable PDF