AP Calculus AB Integration by Substitution Worksheet (PDF)
This u substitution worksheet 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.
These 23 problems practice integration by u-substitution for indefinite and definite integrals, long division, completing the square and choosing a technique (AP Calculus AB, topics 6.9 to 6.14). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.
Later problems use exponential, trig and logarithmic forms, definite integrals with new limits, and situations such as a growing colony and a charging battery. Work in pencil, show every step, and check the answer key only after a full attempt.
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 worksheet: exercises 1 to 10
Exercise 1 – A power of a quadratic
Exercise 2 – A square root
Exercise 3 – Cosine of a multiple
Exercise 4 – Exponential of a multiple
Exercise 5 – Exponential with a quadratic
Find \(\int xe^{x^{2}}\,dx\) and use it to find the shaded area \(\int_{0}^{1}xe^{x^{2}}\,dx\).
Exercise 6 – Sine times a power of cosine
Exercise 7 – Logarithm over x
Exercise 8 – Reciprocal of a linear function
Exercise 9 – Quotient with a quadratic
Exercise 10 – Integral of tangent
Exercises 11 and up
Exercise 11 – Exponential over one plus exponential
Exercise 12 – Secant squared of a multiple
Exercise 13 – Inverse tangent form
Exercise 14 – Cubic inside a power
Exercise 15 – Definite integral with changed limits
Evaluate \(\int_{0}^{2}x(x^{2}+1)^{3}\,dx\) by changing the limits.
Exercise 16 – Definite integral, trig
Exercise 17 – Definite integral, logarithm
Evaluate \(\int_{1}^{e}\frac{\ln x}{x}\,dx\).
Exercise 18 – Definite integral, rational
Exercise 19 – Long division
Exercise 20 – Completing the square
Find \(\int\frac{1}{x^{2}+4x+13}\,dx\).
Exercise 21 – Colony growth
A colony grows at the rate \(r(t)=200te^{-0.1t^{2}}\) bacteria per hour. How many bacteria are added during the first 3 hours?
Exercise 22 – Battery charge
Exercise 23 – Distance from a velocity
Key facts behind this u substitution worksheet
- 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 worksheet
Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.
Download the u substitution worksheet as a PDF with the button below and keep it with your notes.
All AP Calculus AB chapters: AP Calculus AB math.
More AP Calculus AB math
Lessons
See all
Worksheets
See all
Differential Equations and Slope Fields Worksheet- The Chain Rule Worksheet
- Increasing, Decreasing and the First Derivative Test Worksheet
- Power, Sum and Basic Function Derivative Rules Worksheet
- Continuity and the Intermediate Value Theorem Worksheet
- Mean Value Theorem and Extreme Value Theorem Worksheet
Tests
See all- Average Value, Area and Volume Test with Point Values
- Power, Sum and Basic Function Derivative Rules Test with Point Values
- Increasing, Decreasing and the First Derivative Test Test with Point Values
- Optimization Test with Point Values
- The Derivative as a Limit Test with Point Values
- Linearization and L'Hopital's Rule Test with Point Values
