Integrals Evaluate each integral. (4 sech? Vĩ

We have to find definite integral: ∫04sech2 xxdx This will be solved by substitution method or…

Answered: (4 sech? Vĩ

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section: Chapter Questions

Problem 46RE

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Integrals Evaluate each integral.

Expert Solution

Step 1

We have to find definite integral:

$\int_{0}^{4} \frac{s e c h^{2} \sqrt{x}}{\sqrt{x}} d x$

This will be solved by substitution method or change of variable.

Assuming $t = \sqrt{x}$

Limit will change as following,

$\begin{array}{rcl} x & \to & 0 \Rightarrow t \to 0 \\ x & \to & 4 \Rightarrow t \to 2 \end{array}$

Differentiating with respect to $x$ the assumed value, we get

$\begin{array}{rcl} \frac{d t}{d x} & = & \frac{d \sqrt{x}}{d x} \\ \frac{d t}{d x} & = & \frac{1}{2 \sqrt{x}} \\ 2 d t & = & \frac{d x}{\sqrt{x}} \end{array}$

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