integration techniques Archives - Infinity is Really Big

January 27, 2020January 29, 2020

Integration by Parts (and Linear Motion)

Calculus 2, Lectures 3B through 4 (Videotaped Fall 2016)

Integration by parts is needed to compute the definite integral to find the area under this curve representing the speed of a linear motion. — Integration by parts is needed to compute the definite integral to find the area under this curve representing the speed of a linear motion.

You might say that integration by parts is an “unexpected” or “surprising” method of integration.

While the other main method, integration by substitution, can be thought … Read the rest

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