Techniques for Integration: Cheat Sheet

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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 $u$ 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 ( $n$ ) lead to more accurate approximations.

Substitution with Indefinite Integrals
$\displaystyle\int f\left[g(x)\right]{g}^{\prime }(x)dx=\displaystyle\int f(u)du=F(u)+C=F(g(x))+C$
Substitution with Definite Integrals
${\displaystyle\int }_{a}^{b}f(g(x)){g}^{\prime }(x)dx={\displaystyle\int }_{g(a)}^{g(b)}f(u)du$
Integrals of Exponential Functions
$\displaystyle\int {e}^{x}dx={e}^{x}+C$
$\displaystyle\int {a}^{x}dx=\frac{{a}^{x}}{\text{ln}a}+C$
Integration Formulas Involving Logarithmic Functions
$\displaystyle\int {x}^{-1}dx=\text{ln}|x|+C$
$\displaystyle\int \text{ln}xdx=x\text{ln}x-x+C=x(\text{ln}x-1)+C$
$\displaystyle\int {\text{log}}_{a}xdx=\frac{x}{\text{ln}a}(\text{ln}x-1)+C$
Integrals That Produce Inverse Trigonometric Functions
$\displaystyle\int \frac{du}{\sqrt{{a}^{2}-{u}^{2}}}={ \sin }^{-1}\left(\frac{u}{a}\right)+C$
$\displaystyle\int \frac{du}{{a}^{2}+{u}^{2}}=\frac{1}{a}\phantom{\rule{0.05em}{0ex}}{ \tan }^{-1}\left(\frac{u}{a}\right)+C$
$\displaystyle\int \frac{du}{u\sqrt{{u}^{2}-{a}^{2}}}=\frac{1}{a}\phantom{\rule{0.05em}{0ex}}{ \sec }^{-1}\left(\frac{u}{a}\right)+C$

change of variables: the substitution of a variable, such as $u$ , 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

Substitution for Indefinite Integrals

Substitution for Definite Integrals

Integrals of Exponential Functions

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

Approximating Integrals