Chapter 5: Q4P (page 242)

$\int \sqrt{1 + " " a "^{2} x^{2}} d x = \frac{x}{2} \sqrt{1 + " " a "^{2} x^{2}} + \frac{1}{2 a} s i n h^{- 1} \sqrt{a^{2} + b^{2}}$

Short Answer

Expert verified

The required solution is $\int \sqrt{1 + x^{2} a^{2})} d x = \frac{1}{2 a} [\sin h^{- 1} (a x) + 1 + x^{2} a^{2} x a] + c$

Step by step solution

Definition of definite integral

A function that practices the antiderivative of another function is called an indefinite integral. It can be represented graphically as an integral symbol, a function, and finally a dx. The indefinite integral is a more straightforward way of expressing the antiderivative.

Using the substitution

Make use of the substitute $x = (\frac{1}{2}) \sin h u$

role="math" localid="1658836295640" $\int \sqrt{1 + x^{2} a^{2}} d x = \int \sqrt{1 + \sin h^{2} u} \frac{1}{a} \cos h u d u = \frac{1}{a} \int \cos h^{2} u d u = \frac{1}{a} \int \frac{1}{2} (\cos h (2 u) + 1) d u = \frac{1}{2 a} [u + \int \cos h (2 u) d u] = \frac{1}{2 a} [u + \frac{1}{2} \sin h (2 u)] + C = \frac{1}{2 a} [u + \cos h u \sin h u] + C = \frac{1}{2} [\sin h^{- 1} (a x) + \sqrt{1 + \sin h^{2} u x a}] + C = \frac{1}{2} [\sin h^{- 1} (a x) + \sqrt{1 + x^{2} a^{2} x a}] + C$

Now, Apply role="math" localid="1658836375727" $c o s h^{2} " x - s i n h^{2} x = 1 a n d c o s h^{2} x = (1 + c o s h (2 x)) / 2 .$

Therefore, the answer to the question is

$\int \sqrt{1 + x^{2} a^{2}} d x = \frac{1}{2 a} [\sin h^{- 1} (a x) + \sqrt{1 + x^{2} a^{2}} x a] + C$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$\int \sqrt{1 + " " a "^{2} x^{2}} d x = \frac{x}{2} \sqrt{1 + " " a "^{2} x^{2}} + \frac{1}{2 a} s i n h^{- 1} \sqrt{a^{2} + b^{2}}$

Short Answer

Step by step solution

Definition of definite integral

Using the substitution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Linear Momentum

Work Energy and Power

Rotational Dynamics

Quantum Physics

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.

∫1+""a"2x2dx=x21+""a"2x2+12asinh-1a2+b2

Short Answer

Step by step solution

Definition of definite integral

Using the substitution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Linear Momentum

Work Energy and Power

Rotational Dynamics

Quantum Physics

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.

$\int \sqrt{1 + " " a "^{2} x^{2}} d x = \frac{x}{2} \sqrt{1 + " " a "^{2} x^{2}} + \frac{1}{2 a} s i n h^{- 1} \sqrt{a^{2} + b^{2}}$