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Chapter 4: Q. 42 (page 362)

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.
$鈭� \frac{1}{4 + x^{2}} d x$

Short Answer

Expert verified

The solution of the integral is $\frac{1}{2} \tan^{- 1} (\frac{x}{2}) + C .$

Step by step solution

01

Step 1. Given Information.

The given integral is $鈭� \frac{1}{4 + x^{2}} d x .$

02

Step 2. Solve.

By solving the integral we get,

$鈭� \frac{1}{4 + x^{2}} d x = 鈭� \frac{1}{4 (1 + \frac{x^{2}}{4})} d x = \frac{1}{4} 鈭� \frac{1}{1 + {(\frac{x}{2})}^{2}} d x = \frac{1}{4} (2) \tan^{- 1} (\frac{x}{2}) + C = \frac{1}{2} \tan^{- 1} (\frac{x}{2}) + C$

03

Step 3. Verification.

To verify the answer we differentiate $\frac{1}{2} \tan^{- 1} (\frac{x}{2}) + C$ it.

On differentiating we get,

$\frac{1}{2} \tan^{- 1} (\frac{x}{2}) + C = \frac{d}{d x} \frac{1}{2} \tan^{- 1} (\frac{x}{2}) + \frac{d}{d x} C = \frac{1}{2} \frac{d}{d x} \tan^{- 1} (\frac{x}{2}) + \frac{d}{d x} C = \frac{1}{2} 脳 \frac{1}{2} (\frac{1}{1 + {(\frac{x}{2})}^{2}}) + 0 = \frac{1}{4 + x^{2}}$

Hence proved.

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Most popular questions from this chapter

Show by exhibiting a counterexample that, in general, $鈭� \frac{f (x)}{g (x)} d x 鈮� \frac{鈭� f (x) d x}{鈭� g (x) d x}$ . In other words, find two functions f and g such that the integral of their quotient is not equal to the quotient of their integrals.

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} (k^{2} + k + 1)$

If f is negative on [鈭�3, 2], is the definite integral ${鈭�}_{- 3}^{2} f (x) d x$ positive or negative? What about the definite integral 鈭� ${鈭�}_{- 3}^{2} f (x) d x$ ?

Explain why we call the collection of antiderivatives of a function f a family. How are the antiderivatives of a function related?

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. $鈭� (x^{2} - 3 x^{5} - 7) d x$

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