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annie122
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how do u integrate sqrt(1 + x^2) / x??
i reduced this to sec^3(u)/tan(u) but how do i go from here??
i reduced this to sec^3(u)/tan(u) but how do i go from here??
I would be inclined to do this like Random Variable but since you got this far, sec(u)= 1/cos(u) and tan(u)= sin(u)/cos(u) so that [tex]\frac{sec^3(u)}{tan(u)}= \frac{1}{cos^3(u)}\frac{cos(u)}{sin(u)}= \frac{1}{sin(u)cos^2(u)}[/tex]which has sine to an odd power. We can multiply both numerator and denominator by sin(x) to get [tex]\frac{sin(x)}{sin^2(x)cos^2(x)}= \frac{sin(x)}{(1- cos^2(x))cos^2(x)}[/tex]Yuuki said:how do u integrate sqrt(1 + x^2) / x??
i reduced this to sec^3(u)/tan(u) but how do i go from here??
[tex]\int \frac{\sqrt{1 + x^2}}{x}\,dx[/tex]
[tex]\text{Let }\,x \,=\,\tan\theta \quad\Rightarrow\quad dx \,=\,\sec^2\!\theta\,d\theta[/tex]
[tex]\text{I reduced this to: }\:\int \frac{\sec^3\!\theta}{\tan\theta}\,d\theta[/tex] . Good!
Trig substitution is a method used in calculus to simplify integrals involving expressions with radicals. It involves substituting a trigonometric function for a variable in the integral.
In this specific integral, using trig substitution is necessary because the expression inside the radical cannot be simplified using algebraic methods. Trig substitution allows us to rewrite the integral in terms of a trigonometric function that can be easily integrated.
The choice of trigonometric function to substitute depends on the form of the expression inside the radical. In this integral, we use the substitution x = tan(θ) because it will eliminate the radical and result in a simpler integral.
Yes, there are various other substitution methods in calculus, such as u-substitution and integration by parts. However, in this specific integral, trig substitution is the most efficient method.
The square root in the expression is used to simplify the integral and make it easier to integrate. Without the square root, the integral would be more complex and difficult to solve.