Book cover for Calculus

Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

ISBN #9781119379331

7th Edition

5,101 Questions

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33,688 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Integration by substitution is a powerful tool that reverses the chain rule in differentiation. By identifying an 'inside function' and its derivative within the integrand, one can simplify complex integrals into standard forms. This method applies to both indefinite and definite integrals, though care must be taken to adjust for any constant factors or mismatches when the derivative does not perfectly match the required form.

Learning Objectives

1

Explain the method of integration by substitution and its theoretical basis using the chain rule.

2

Identify the 'inside function' and its derivative within an integrand to apply an appropriate substitution.

3

Solve both indefinite and definite integrals by converting them into standard forms through suitable substitutions.

4

Recognize when substitution is applicable and how to adjust for missing constant factors during antiderivatives.

Key Concepts

CONCEPT

DEFINITION

Integration by Substitution

A technique for finding antiderivatives by replacing a composite function with a new variable, often simplifying the integral into a standard form.

Inside Function (u-substitution)

The function chosen to substitute (often denoted as w or u) whose derivative is also present in the integrand up to a constant factor.

Chain Rule

A rule in differentiation stating that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Its reverse underpins the substitution method.

Antiderivative

A function whose derivative is the given function. In substitution, we seek an antiderivative expressed in terms of the new variable before converting back to the original variable.

Example Problems

Example 1

Use substitution to express each of the following integrals as a multiple of $\int_{a}^{b}(1 / w) d w$ for some $a$ and $b$ Then evaluate the integrals. (a) $\int_{0}^{1} \frac{x}{1+x^{2}} d x$ (b) $\int_{0}^{\pi / 4} \frac{\sin x}{\cos x} d x$

Example 2

(a) Find the derivatives of $\sin \left(x^{2}+1\right)$ and $\sin \left(x^{3}+1\right)$ (b) Use your answer to purt (a) to find antiderivatives of: (i) $x \cos \left(x^{2}+1\right)$ (ii) $x^{2} \cos \left(x^{3}+1\right)$ (c) Find the general antiderivatives of: (i) $x \sin \left(x^{2}+1\right)$ (ii) $x^{2} \sin \left(x^{3}+1\right)$

Example 3

Find the integrals Check your answers by differentiation. $$\int e^{3 x} d x$$

Example 4

Find the integrals Check your answers by differentiation. $$\int t e^{t^{2}} d t$$

Example 5

Find the integrals Check your answers by differentiation. $$\int e^{-x} d x$$
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Step-by-Step Explanations

QUESTION

How do we evaluate ∫ x^3√(x^4+5) dx using substitution?

STEP-BY-STEP ANSWER:

Step 1: Identify the inside function. Let w = x^4 + 5.
Step 2: Differentiate w to obtain dw/dx = 4x^3. Hence, dw = 4x^3 dx or x^3 dx = (1/4) dw.
Step 3: Substitute into the integral. The integral becomes (1/4) ∫ √(w) dw.
Step 4: Rewrite √(w) as w^(1/2) and integrate: ∫ w^(1/2) dw = (2/3)w^(3/2) + C.
Step 5: Multiply by the constant: (1/4)*(2/3)w^(3/2) + C = (1/6)w^(3/2) + C.
Step 6: Substitute back w = x^4 + 5, yielding the final answer: (1/6)(x^4+5)^(3/2) + C.
Final Answer: (1/6)(x^4+5)^(3/2) + C.

Example: ∫ x^3√(x^4+5) dx

QUESTION

How do we evaluate ∫ 3x^2 cos(x^3) dx using substitution?

STEP-BY-STEP ANSWER:

Step 1: Identify the inside function. Let w = x^3.
Step 2: Differentiate w to obtain dw/dx = 3x^2, meaning dw = 3x^2 dx.
Step 3: Substitute into the integral. The integral becomes ∫ cos(w) dw.
Step 4: Integrate cos(w) with respect to w: ∫ cos(w) dw = sin(w) + C.
Step 5: Replace w with x^3, yielding the final answer: sin(x^3) + C.
Final Answer: sin(x^3) + C.

Example: ∫ 3x^2 cos(x^3) dx

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Common Mistakes

  • Failing to include the constant factor when the derivative of the chosen substitution does not exactly match a factor in the integrand.
  • Not adjusting the limits of integration in definite integrals when switching variables.
  • Mixing up the roles of the inside function and its derivative, which might lead to an incorrect substitution.
  • Overcomplicating the substitution by choosing a function that does not simplify the integrand effectively.