- #1
karush
Gold Member
MHB
- 3,269
- 5
$I=\int\frac{1}{\sqrt{1+x^2}}$
I followed this example but it surprised me
$x = \tan\theta \ \ dx = \sec^{2}\theta{d\theta}$
$$I=\int\left(\frac{1}{\sec\theta}\right){\sec^{2}\theta{d\theta}}
=\int{\sec\theta{d\theta}}$$
Why does this need to be done?
$$I=\int{\left(\frac{\sec^2\theta + \sec\theta\tan\theta}{\sec\theta + \tan\theta}\right){d\theta}}$$
Let, $(\sec\theta + \tan\theta) = u$
$(\sec^2\theta + \sec\theta\tan\theta)d\theta = du$
$$I=\int\frac{du}{u}$$
$$I=\ln{u}+c$$
$$I=\ln{\vert\sec\theta + \tan\theta\vert}+c$$
$$I=\ln{\vert\sqrt{1+\tan^2\theta} + \tan\theta\vert}+c$$
$I=\ln{\vert\sqrt{1+x^2} + x\vert}+c$, where $c$ is a constant
I followed this example but it surprised me
$x = \tan\theta \ \ dx = \sec^{2}\theta{d\theta}$
$$I=\int\left(\frac{1}{\sec\theta}\right){\sec^{2}\theta{d\theta}}
=\int{\sec\theta{d\theta}}$$
Why does this need to be done?
$$I=\int{\left(\frac{\sec^2\theta + \sec\theta\tan\theta}{\sec\theta + \tan\theta}\right){d\theta}}$$
Let, $(\sec\theta + \tan\theta) = u$
$(\sec^2\theta + \sec\theta\tan\theta)d\theta = du$
$$I=\int\frac{du}{u}$$
$$I=\ln{u}+c$$
$$I=\ln{\vert\sec\theta + \tan\theta\vert}+c$$
$$I=\ln{\vert\sqrt{1+\tan^2\theta} + \tan\theta\vert}+c$$
$I=\ln{\vert\sqrt{1+x^2} + x\vert}+c$, where $c$ is a constant
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