\int_{\frac{1}{120}}^{\frac{1}{60}}(120\pi cos(120\pi t)dt u = 120 \pi t du = lower limit, in u =...


d. {eq}\int_{\frac{1}{120}}^{\frac{1}{60}}(120\pi cos(120\pi t)dt {/eq} {eq}u {/eq} = {eq}120 {/eq}{eq}\pi t {/eq}

du =

lower limit, in u =

upper limit, in u =

rewrite the integral and find the answer using u-variable

Integration by Substitution:

It is the method of integration where variables like u or v are substituted to simplify the given complex function. For example, {eq}\displaystyle \int p(q(y)) q'(y) y = P(q(y))+C. {/eq}

Where C is the constant of the integration.

The more useful and important applications:

1. Move the constant out: {eq}\displaystyle \int a\cdot f\left(w\right)dw=a\cdot \int f\left(w\right)dw. {/eq}

2. Common integration: {eq}\displaystyle \int \cos \left(u\right)du=\sin \left(u\right). {/eq}

3. Value of {eq}\displaystyle \sin \left(2\pi \right) = 0. {/eq}

4. Value of {eq}\displaystyle \sin \left(\pi \right) = 0. {/eq}

Answer and Explanation:

The task is to solve the integration of Value of $$\displaystyle I = \int_{\frac{1}{120}}^{\frac{1}{60}}(120\pi cos(120\pi t)dt $$

Move the constant out.

$$\displaystyle = 120\pi \cdot \int _{\frac{1}{120}}^{\frac{1}{60}}\cos \left(120\pi t\right)dt $$

Apply the substitution for {eq}u=120\pi t \Rightarrow du = 120\pi dt. {/eq}

Limits: {eq}\frac{1}{120} \rightarrow \pi {/eq} and {eq}\frac{1}{60} \rightarrow 2\pi. {/eq}

$$\displaystyle = 120\pi \cdot \int _{\pi }^{2\pi }\cos \left(u\right)\frac{1}{120\pi }du $$

Move the constant out.

$$\displaystyle = 120\pi \frac{1}{120\pi }\cdot \int _{\pi }^{2\pi }\cos \left(u\right)du $$

Use the common integration.

$$\displaystyle = 120\pi \frac{1}{120\pi }\left[\sin \left(u\right)\right]^{2\pi }_{\pi } $$


$$\displaystyle = \left[\sin \left(u\right)\right]^{2\pi }_{\pi } $$

Compute the boundaries.

$$\begin{align*} \displaystyle &= 0 - 0\\ \displaystyle &= 0. \end{align*} $$

Learn more about this topic:

How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5

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