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“Face Your Fears and Run Towards The Danger”
U-substitution is the first integration technique that should be considered before pursuing the implementation of a more advanced approach. This technique, which is analogous to the chain rule of differentiation, is useful whenever a function composition can be found within the integrated.
\(\quad\displaystyle\int f’\big(g(x)\big)g'(x) = f\big(g(x)\big)+C\)
The main objective of u-substitution is to express a given integral as another integral, preferably as an integral whose antiderivative is readily known. See Integral Formulas.
After deciding upon the substitution that is needed to reexpress an integral in terms of \(x\) into terms of \(u\), it is crucial to also express \(\boldsymbol dx\) into terms of \(du\). Failing to do so can make the problem unbearably difficult to integrate. DO NOT forgot to convert the \(\boldsymbol dx\).
One of the difficulties associated with this technique involves making a strategic choice for the substitution \(u:=u(x)\). To help with this decision process, listed below are some suggestive strategies that you might wish to consider. It should be noted that this list is not exhaustive and you may opt to use a different approach.
| Some Suggestive Guidelines |
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When applying u-substitution you MUST convert EVERYTHING over.
A “good” substitution is one that leaves no \(x\)’s in the converted integral. Remember that you need to ALWAYS convert (1) the integrated and (2) the \(dx\), and if the integral is a definite integral then (3) also express the endpoints of integration in terms of \(x\)
After integrating the converted integral \(\int g(u)du\) besure to convert back to the original variable. Because we are not solving for \(u\) rather for \(x\).
Integration by parts is an integration technique that involves separating the integrand into two parts. This technique is useful whenever a product appears in the integrated, and the implement u-substitution is not feasible.
\(\displaystyle\int u\,dv = uv\,-\!\int v\,du\)
Make a choice for \(u:=u(x)\), which forces you into a choice for \(dv\). Then, compute the differential \(du=\frac{d}{dx}u\) and find \(v(x)\) by integrating \(v=\int dv\).
When determining what function to set as \(u:=u(x)\) in this procedure, you may wish to consider the following suggestions.
Note: U-Substitution focuses on simplifying by reducing composition by subbing out functions. Integration By Parts can be seen as a two-fold substitution procedure that focuses on products of function NOT compositions.
When analytical methods fail to yield a result, a numerical approximation is arguably the next best answer that can be provided. We begin by partitioning interval \([a,b]\) into \(n\) subintervals. For computational purposes, it is customary to subdivide the given interval equally into \(n\) subintervals.
\(\displaystyle\Delta x=\frac{\text{length of interval}}{\text{number of pieces}}=\frac{b-a}{n}\)
| Method | Formula |
| Left-Hand Rule | \(\displaystyle\texttt{Area}\approx \sum_{n=0}^{n-1}f(x_k)\Delta x\) |
| Right-Hand Rule | \(\displaystyle\texttt{Area}\approx\sum_{k=1}^{n}f(x_k)\Delta x\) |
| Midpoint Rule | \(\displaystyle\sum_{k=1}^{n}f(\overline{x}_k)\Delta x\) |
| Trapezoidal Rule | \(\displaystyle T_n=\frac{b-a}{2n}\big(f(x_0)+2f(x_1)+\cdots+2f(x_{n+-1})+f(x_n) \big)\) |
| Simpson’s \(\frac{3}{8}\) Rule | \(\displaystyle S_n=\frac{b-a}{3n}\big( f(x_0)+4f(x_1)+2f(x_2)+\cdots+4f(x_{n-1})+2f(x_n) \big)\) |
| Method | Error Term |
| Left-Hand/Right-Hand Rule | |
| Midpoint Rule | \(\displaystyle|E_T|\leq\frac{M_2(b-a)^3}{24n^4}\) |
| Trapezoidal Rule | \(\displaystyle|E_T|\leq\frac{M_2(b-a)^3}{12n^4}\) |
| Simpson’s \(\frac{3}{8}\) Rule | \(\displaystyle|E_S|\leq\frac{M_4(b-a)^5}{180n^4}\) |
When integrating products of trigonometric functions, the general practice involves applying the trigonometric versions of the Pythagorean Theorem such as \(\texttt{sin}^2(\theta)+\texttt{cos}^2(\theta)=1\) or \(\texttt{tan}^2(\theta)+1=\texttt{sec}^2(\theta)\) in conjunction with an appropriate u-substitution.
\(\int \texttt{sin}^{n}(\theta)\texttt{cos}^{m}(\theta)\, d\theta\) |
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\(\int\texttt{tan}^{n}{(\theta)}\texttt{sec}^{m}{(\theta)}\, d\theta\) |
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This integration technique is particularly useful whenever a sum-of-square or a difference-of-squares is found within the integrand. This approach is typically implemented to help reduce the complexity found among square-root functions.
These problems often require using one of the Trigonometric versions of the Pythagorean Theorem.
| Difference of Squares | ||
| Substitution | Expression | Protocols |
| Sine | \(\sqrt{a^2-u^2}\) | \(u=a\,\texttt{sin}(\theta)\) |
| Secant | \(\sqrt{u^2-a^2}\) | \(u=a\,\texttt{sec}(\theta)\) |
| Sum of Squares | ||
| Substitution | Expression | Protocols |
| Tangent | \(\sqrt{a^2+u^2}\) | \(u=a\,\texttt{tan}(\theta)\) |
Although partial fraction decomposition (or simply partial fractions) is an algebraic, not an integration technique, it is useful for integrating rational functions. When decomposing the ration function, remember to echo any repeated factors. A rational functions has the form \(\textstyle f(x)=\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are both polynomials.
The following functions are examples of rational functions.
\(\frac{2x}{x^2+5x+6}\), \(\frac{x^2+2x+2}{x^3+6x^28x}\), \(\frac{x^3+2x^2+4x+3}{x^2-5x+14}\).
Whereas the following functions are NOT rational functions.
\(\frac{x+1}{x^2+5\sqrt{x}+6}\), \(\frac{\sqrt{x^2-x+6}}{x^3+6x^28x}\), \(\frac{x^3+2x^2+4x+3}{x^2-5x+14}\)
Whereas the following function IS a rational function but the degree of the numerator exceeds the degree of the denominator.
\(\frac{x^3+2x^2+4x+3}{x^2-5x+14}\)
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| Type 1: Unbounded Regions |
|
| Type 2: Discontinuities |
|
Common Errors
\( \displaystyle \int_{0}^{a} \frac{dx}{x^p}=\left\{\begin{array}{ll} \text{Divergence}& p\geq1\\ \displaystyle \frac{a^{p-1}}{1-p} & p<1\end{array}\right. \)
\( \displaystyle \int_{a}^{\infty} \frac{dx}{x^p}=\left\{\begin{array}{ll} \text{Divergence}& p\leq1\\ \displaystyle \frac{a^{p-1}}{p-1} & p>1\end{array}\right. \)
U-substitution is the first integration technique that should be considered before pursuing the implementation of a more advanced approach. This technique, which is analogous to the chain rule of differentiation, is useful whenever a function composition can be found within the integrand.
\(\quad\displaystyle\int f’\big(g(x)\big)g'(x) = f\big(g(x)\big)+C\)Integration by parts is an integration technique that involves separating the integrand into two parts. This technique is useful whenever a product appears in the integrated and the implement u-substitution is not feasible.
\(\displaystyle\int u\,dv = uv\,-\!\int v\,du\)When integrating products of trigonometric functions, the general practice involves applying the trignometric versions of the Pythagorean Theorem such as
\(\texttt{sin}^2(\theta)+\texttt{cos}^2(\theta)=1\) or \(\texttt{tan}^2(\theta)+1=\texttt{sec}^2(\theta)\) in conjunction with an appropriate u-substitution.
This integration technique is particularly useful whenever either a sum-of-square or a difference-of-squares is found within the integrand. This approach is typically implemented to help reduce the complexity found among square-root functions.
Difference of Squares
| Substitution | Expression | Protocols |
| Sine | \(\sqrt{a^2-u^2}\) | \(u=a\,\texttt{sin}(\theta)\) |
| Secant | \(\sqrt{x^2-a^2}\) | \(u=a\,\texttt{sec}(\theta)\) |
Sum of Squares
| Substitution | Expression | Protocols |
| Tangent | \(\sqrt{a^2+u^2}\) | \(u=a\,\texttt{tan}(\theta)\) |
* Note: The presence of the square-root function is not required to implement these techniques.
Although partial fraction decomposition (or simply partial fractions) is an algebraic, not an integration technique, it is useful for integrating rational functions. When decomposing the ration function, remember to echo any repeated factors.
A rational functions has the form \(f(x)=\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are both polynomials.
The following functions are examples of rational functions. \(\frac{2x}{x^2+5x+6}\), \(\frac{x^2+2x+2}{x^3+6x^28x}\), \(\frac{x^3+2x^2+4x+3}{x^2-5x+14}\)
The following functions are examples of functions that are NOT rational functions. \(\frac{x+1}{x^2+5\sqrt{x}+6}\), \(\frac{\sqrt{x^2-x+6}}{x^3+6x^28x}\), \(\frac{x^3+2x^2+4x+3}{x^2-5x+14}\)
Decomposition Steps
Type 1: Unbounded Regions
Type 2: Discontinuities
Common Errors
\(\textstyle\int_{-\infty}^{\infty} f(x)\,dx\neq\underset{t\to\infty}{\lim}\int_{-t}^{t} f(x)\,dx \)
When analytical methods fail to yield a result, a numerical approximation is arguably the next best answer that can be provided. We begin by partitioning interval \([a,b]\) into \(n\) subintervals . For compuational purposes its is customary to subdivide the equally into \(n\) subintervals.
\(\displaystyle\Delta x=\frac{\text{length of interval}}{\text{number of pieces}}=\frac{b-a}{n}\)
| Method | Formula |
| Left-Hand Rule | |
| Right-Hand Rule | |
| Midpoint Rule | |
| Trapezoidal Rule | |
| Simpson’s \(\frac{3}{8}\) Rule |
| Method | Error Term |
| Left-Hand/Right-Hand Rule | |
| Midpoint Rule | |
| Trapezoidal Rule | |
| Simpson’s \(\frac{3}{8}\) Rule |
U-substitution is the first integration technique that should be considered before pursuing the implementation of a more advanced approach. This technique, which is analogous to the chain rule of differentiation, is useful whenever a function composition can be found within the integrand.
\(\quad\displaystyle\int f’\big(g(x)\big)g'(x) = f\big(g(x)\big)+C\)Integration by parts is an integration technique that involves separating the integrand into two parts. This technique is useful whenever a product appears in the integrated and the implement u-substitution is not feasible.
\(\displaystyle\int u\,dv = uv\,-\!\int v\,du\)When analytical methods fail to yield a result, a numerical approximation is arguably the next best answer that can be provided. We begin by partitioning interval \([a,b]\) into \(n\) subintervals. For computational purposes, it is customary to subdivide the given interval equally into \(n\) subintervals.
\(\displaystyle\Delta x=\frac{\text{length of interval}}{\text{number of pieces}}=\frac{b-a}{n}\)
| Method | Formula |
| Left-Hand Rule | |
| Right-Hand Rule | |
| Midpoint Rule | |
| Trapezoidal Rule | |
| Simpson’s \(\frac{3}{8}\) Rule |
| Method | Error Term |
| Left-Hand/Right-Hand Rule | |
| Midpoint Rule | |
| Trapezoidal Rule | |
| Simpson’s \(\frac{3}{8}\) Rule |
When integrating products of trigonometric functions, the general practice involves applying the trignometric versions of the Pythagorean Theorem such as \(\texttt{sin}^2(\theta)+\texttt{cos}^2(\theta)=1\) or \(\texttt{tan}^2(\theta)+1=\texttt{sec}^2(\theta)\) in conjunction with an appropriate u-substitution.
This integration technique is particularly useful whenever either a sum-of-square or a difference-of-squares is found within the integrand. This approach is typically implemented to help reduce the complexity found among square-root functions.
Difference of Squares
| Substitution | Expression | Protocols |
| Sine | \(\sqrt{a^2-u^2}\) | \(u=a\,\texttt{sin}(\theta)\) |
| Secant | \(\sqrt{x^2-a^2}\) | \(u=a\,\texttt{sec}(\theta)\) |
Sum of Squares
| Substitution | Expression | Protocols |
| Tangent | \(\sqrt{a^2+u^2}\) | \(u=a\,\texttt{tan}(\theta)\) |
Although partial fraction decomposition (or simply partial fractions) is an algebraic, not an integration technique, it is useful for integrating rational functions. When decomposing the ration function, remember to echo any repeated factors.
A rational functions has the form \(f(x)=\frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are both polynomials.
The following functions are examples of rational functions. \(\frac{2x}{x^2+5x+6}\), \(\frac{x^2+2x+2}{x^3+6x^28x}\), \(\frac{x^3+2x^2+4x+3}{x^2-5x+14}\)
The following functions are examples of functions that are NOT rational functions. \(\frac{x+1}{x^2+5\sqrt{x}+6}\), \(\frac{\sqrt{x^2-x+6}}{x^3+6x^28x}\), \(\frac{x^3+2x^2+4x+3}{x^2-5x+14}\)
Decomposition StepsCheck with your tutor
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