Answered: Find the indefinite integral in two…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter9: Multivariable Calculus

Section9.CR: Chapter 9 Review

Problem 57CR

See similar textbooks

Related questions

Question

Find the indefinite integral in two different ways, (a) use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant, and (b) verify analytically that the results differ only by a constant. ∫sec² x tan x dx

Expert Solution

Step 1: Given

The indefinite integral, $\int s e c^{2} (x) \tan (x) d x$ .

Step 2: To determine

To find the indefinite integral in two different ways to show that the results differ only by a constant:

(a) By using the graph and (b) By Analytically.

(b) By Analytically.

Let us consider, $\int s e c^{2} (x) \tan (x) d x$ .

Put $u = \tan (x) \Rightarrow d u = s e c^{2} (x) d x$ .

$\begin{array}{rcl} ∴ \int s e c^{2} (x) \tan (x) d x & = & \int u d u \\ = & \frac{u^{2}}{2} + C - - - \{∵ \int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C\} \\ = & \frac{\tan^{2} (x)}{2} + C - - \{∵ u = \tan (x)\} \end{array}$

Step 3

Therefore,

$\begin{array}{rcl} ∴ \int s e c^{2} (x) t a n (x) d x & = & \frac{t a n^{2} (x)}{2} + C \end{array}$ ,

Where, C is any constant.

If we take C=1 then the value of the indefinite integral will be different as compared to if we take C=(-1) or any other constant.

Step by step

Solved in 5 steps with 1 images

Knowledge Booster

$Indefinite Integral$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.