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Chapter 5: Q. 70 (page 465)

Solve each of the integrals in Exercises $39 鈥� 74$ . Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$鈭� \ln (4 + x^{2}) d x$

Short Answer

Expert verified

The value of integral is $x \ln (x^{2} + 4) - 2 x + 4 \tan^{- 1} (\frac{x}{2}) + C$

Step by step solution

01

Step 1. Given information

An expression is given as $鈭� \ln (4 + x^{2}) d x$

02

Step 2. Solving integral

Let

$u = \ln (x^{2} + 4) d u = \frac{2 x}{x^{2} + 4} d x v = 鈭� 1 d x = x$

Now substitute the value and simplify as,

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

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