substitution Archives - Infinity is Really Big

January 23, 2020January 23, 2020

Integration by Substitution (Method of Integration)

Calculus 2, Lectures 2A through 3A (Videotaped Fall 2016)

How to visualize integration by substitution for a definite integral. The starting and ending areas are the same. — The integral $\displaystyle\int_{0}^{\sqrt{\pi/2}}2x\cos(x^{2})\, dx$ gets transformed to the integral $\displaystyle\int_{0}^{\pi/2}\cos(u)\, du$ under the substitution $u=x^{2}$ and $du=2xdx$ .

\displaystyle\int_{0}^{\sqrt{\pi/2}}2x\cos(x^{2})\, dx — The integral $\displaystyle\int_{0}^{\sqrt{\pi/2}}2x\cos(x^{2})\, dx$ gets transformed to the integral $\displaystyle\int_{0}^{\pi/2}\cos(u)\, du$ under the substitution $u=x^{2}$ and $du=2xdx$ .

In Calculus 1, the techniques of integration introduced are usually pretty straightforward. In fact, they are usually just memorized as basic facts about antiderivatives.

For Calculus 2, various new integration techniques are introduced, including integration by … Read the rest

May 30, 2019September 5, 2019

Systems of Linear Equations in Two Dimensions

Visual Linear Algebra Online, Section 1.3

Visualizing the parametric vector form of the solution set of a system with infinitely many solutions.

Jack and Jill earn a combined $79.50 when Jack works 2 hours and Jill works 3 hours. On the other hand, if Jack works 3 hours and Jill works 2 hours, they make $78.75 total. What are

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January 10, 2019July 4, 2019

Financial Mathematics Tools: Substitutions, Quadratic Formula, Logarithms, and “Functional-Thinking”

The quadratic formula

Useful pre-university mathematics is not restricted to arithmetic, percentages, and basic algebra.

Functional-thinking is one key to doing many other useful things. Tools such as substitutions, the quadratic formula, and logarithms also come in handy.

These tools are used in engineering and science. For example, in the evaluation of useful integrals (such as those that come up in … Read the rest