AP Calculus AB Integration by Substitution Test with Point Values (PDF)

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

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This u substitution test 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.

This test checks integration by u-substitution for indefinite and definite integrals, long division, completing the square and choosing a technique (AP Calculus AB, 6.9 to 6.14). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.

Time and points

About this test

Grade: AP Calculus AB
Chapter: Integration by Substitution
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
Time: 45 minutes
Total: 20 points

U substitution test: questions

Question 1 – Indefinite substitution [2 pts]

Find \(\int 3x^{2}(x^{3}-5)^{4}\,dx\).

Question 2 – Definite substitution [3 pts]

The graph of f(x) = 2x e^(x²).

Evaluate \(\int_{0}^{2}2xe^{x^{2}}\,dx\) (the shaded area). Show the change of limits.

Question 3 – Exponential substitution [2 pts]

Find \(\int e^{3x+1}\,dx\).

Question 4 – Trigonometric substitution [3 pts]

Evaluate \(\int_{0}^{\pi/2}\cos x\sin^{3}x\,dx\).

Question 5 – Logarithm form [3 pts]

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

Evaluate \(\int_{1}^{e}\frac{1}{x(1+\ln x)}\,dx\).

Question 6 – Long division [2 pts]

Find \(\int\frac{x^{2}+5}{x+2}\,dx\).

Question 7 – Inflow of water [3 pts]

Inflow rate in gallons per minute.

Water flows into a tank at \(r(t)=40te^{-t^{2}/10}\) gallons per minute. (a) Write an integral for the water added in the first 5 minutes. (1 point) (b) Evaluate it exactly. (2 points)

Question 8 – Choosing a technique [2 pts]

Name the technique and give the antiderivative: (a) \(\int x\cos(x^{2})\,dx\) (b) \(\int\frac{x^{2}+1}{x}\,dx\). (1 point each)

Key facts behind this u substitution test

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

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All AP Calculus AB chapters: AP Calculus AB math.

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