Evaluate the integral: \int \frac{1}{(a (\sqrt{4x^2 + 1})} dx


Evaluate the integral:

{eq}\int \frac{1}{a \sqrt{4x^2 + 1}} dx{/eq}

Indefinite Integral:

If {eq}F(x) {/eq} is anti-derivative of {eq}f(x) {/eq}, then anti-derivative of {eq}f(x) {/eq} is called an indefinite integral and denoted by,

{eq}\int {f(x)dx = F(x) + c} {/eq}, where c is constant of integration.

It is also known as anti-differentiation i.e. reversing procedure of differentiation.

We use the following formula in the given problem:

{eq}\eqalign{ & \frac{d}{{dx}}\left( {\tan x} \right) = {\sec ^2}x \cr & \int {\sec xdx} = \ln \left| {\sec x + \tan x} \right| + c, \cr} {/eq}

where {eq}c {/eq} is arbitrary constant.

Answer and Explanation:

Let {eq}I = \int \frac{1}{a \sqrt{4x^2 + 1}} dx, {/eq}

Substitute {eq}x = \frac{1}{2}\tan t \Rightarrow t = {\tan ^{ - 1}}2x, {/eq} and {eq}dx =...

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Learn more about this topic:

Indefinite Integral: Definition, Rules & Examples

from Calculus: Tutoring Solution

Chapter 7 / Lesson 14

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