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Essential Concepts
Integration using Substitution
- Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term ‘substitution’ refers to changing variables or substituting the variable [latex]u[/latex] and du for appropriate expressions in the integrand.
- When using substitution for a definite integral, we also have to change the limits of integration.
Integrals Involving Exponential and Logarithmic Functions
- Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.
- Substitution is often used to evaluate integrals involving exponential functions or logarithms.
Integrals Resulting in Inverse Trigonometric Functions
- Formulas for derivatives of inverse trigonometric functions developed in Derivatives of Exponential and Logarithmic Functions lead directly to integration formulas involving inverse trigonometric functions.
- Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem.
- Substitution is often required to put the integrand in the correct form.
Approximating Integrals
- All differentiable functions can have derivatives found using established calculus rules.
- Not all functions can be integrated into a simple antiderivative form using elementary functions.
- For functions that do not have a straightforward antiderivative, integration can be approximated using methods such as Riemann sums.
- Riemann sums approximate the area under a curve by dividing the area into rectangles and summing their areas.
- When approximating integrals, providing upper and lower bounds helps determine the range within which the true value lies.
- Upper sums use the maximum function value on each subinterval, while lower sums use the minimum function value.
- The interval between the upper and lower sums gives an estimate of the possible error in the approximation.
- Larger numbers of subintervals ([latex]n[/latex]) lead to more accurate approximations.
Key Equations
- Substitution with Indefinite Integrals
[latex]\displaystyle\int f\left[g(x)\right]{g}^{\prime }(x)dx=\displaystyle\int f(u)du=F(u)+C=F(g(x))+C[/latex] - Substitution with Definite Integrals
[latex]{\displaystyle\int }_{a}^{b}f(g(x)){g}^{\prime }(x)dx={\displaystyle\int }_{g(a)}^{g(b)}f(u)du[/latex] - Integrals of Exponential Functions
[latex]\displaystyle\int {e}^{x}dx={e}^{x}+C[/latex]
[latex]\displaystyle\int {a}^{x}dx=\frac{{a}^{x}}{\text{ln}a}+C[/latex] - Integration Formulas Involving Logarithmic Functions
[latex]\displaystyle\int {x}^{-1}dx=\text{ln}|x|+C[/latex]
[latex]\displaystyle\int \text{ln}xdx=x\text{ln}x-x+C=x(\text{ln}x-1)+C[/latex]
[latex]\displaystyle\int {\text{log}}_{a}xdx=\frac{x}{\text{ln}a}(\text{ln}x-1)+C[/latex] - Integrals That Produce Inverse Trigonometric Functions
[latex]\displaystyle\int \frac{du}{\sqrt{{a}^{2}-{u}^{2}}}={ \sin }^{-1}\left(\frac{u}{a}\right)+C[/latex]
[latex]\displaystyle\int \frac{du}{{a}^{2}+{u}^{2}}=\frac{1}{a}\phantom{\rule{0.05em}{0ex}}{ \tan }^{-1}\left(\frac{u}{a}\right)+C[/latex]
[latex]\displaystyle\int \frac{du}{u\sqrt{{u}^{2}-{a}^{2}}}=\frac{1}{a}\phantom{\rule{0.05em}{0ex}}{ \sec }^{-1}\left(\frac{u}{a}\right)+C[/latex]
Glossary
- change of variables
- the substitution of a variable, such as [latex]u[/latex], for an expression in the integrand
- integration by substitution
- a technique for integration that allows integration of functions that are the result of a chain-rule derivative
Study Tips
Substitution for Indefinite Integrals
- Practice recognizing integrands in the form [latex]f(g(x))g'(x)[/latex]
- Remember that [latex]du = g'(x)dx[/latex]
- Don’t forget to substitute back to [latex]x[/latex] at the end
- Be prepared to adjust constants or solve for [latex]x[/latex] in terms of [latex]u[/latex]
- If the substitution doesn’t work, try a different [latex]u[/latex]
Substitution for Definite Integrals
- Practice changing limits of integration when substituting
- Remember to adjust for any constant factors when substituting
- Look for opportunities to simplify the integrand before substituting
- Check your answer by differentiating the result
Integrals of Exponential Functions
- Practice identifying when to use direct formula vs. substitution
- Remember that [latex]e^x[/latex] is its own derivative, which simplifies many integrals
- For definite integrals, be comfortable with changing limits of integration
Integrals Involving Logarithmic Functions
- Practice identifying when to use direct formula vs. substitution
- Remember the relationship between exponential and logarithmic functions
- Practice integrating a variety of logarithmic expressions and reciprocal functions
- Review properties of logarithms to simplify complex expressions before integration
Integrals Resulting in Inverse Trigonometric Functions
- Remember domain restrictions for inverse trig functions
- Review trigonometric identities
- Practice both indefinite and definite integrals
Approximating Integrals
- Practice identifying functions without elementary antiderivatives
- Understand when to use left-endpoint vs. right-endpoint sums
- Remember that increasing [latex]n[/latex] generally improves approximation accuracy
- Be comfortable with calculating and interpreting error bounds
- Visualize the approximation process using graphs