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Chapter 5: Q. 56 (page 465)

Solve the integral
$鈭� \frac{1}{\sqrt{x^{2} + 9}} d x$

Short Answer

Expert verified

$\sin h^{- 1} (\frac{x}{3}) + c$

Step by step solution

01

Step 1. Given

The given integral is $鈭� \frac{1}{\sqrt{x^{2} + 9}} d x$ .

02

Step 2. Simplifying the integral using identity

$To solve the integral, le t x = 3 s i n h (u) . Now, derivation of the above equation is solved below . x = 3 s i n h (u) . d x = c o s h (u) d u Now, use the identity s i n h^{2} (u) + 1 = c o s h^{2} (u) 鈭� \frac{1}{\sqrt{x^{2} + 9}} d x = 鈭� \frac{1}{\sqrt{(3 s i n h (u) .^{2}) + 9}} d x = 鈭� \frac{1}{3 \sqrt{c o s h^{2} (u)}} d x = 鈭� \frac{1}{\cos h (u)}$

03

Step 3. Solving the integral

$鈭� \frac{1}{\sqrt{x^{2} + 9}} d x = 鈭� 1 d u = u = \sin h^{- 1} (\frac{x}{3}) + c$

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Most popular questions from this chapter

Find three integrals in Exercises 21鈥�70 that we can anti-differentiate immediately after algebraic simplification.

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