5.4 Indefinite Integrals and the Net Change Theorem/37: Difference between revisions
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&= \int_{0}^{\frac{\pi}{4}}\left(\frac{1}{\cos^2(\theta)} + \frac{\cos^2(\theta)}{\cos^2(\theta)}\right)d\theta | &= \int_{0}^{\frac{\pi}{4}}\left(\frac{1}{\cos^2(\theta)} + \frac{\cos^2(\theta)}{\cos^2(\theta)}\right)d\theta | ||
= \int_{0}^{\frac{\pi}{4}}\left(\sec^2(\theta) + 1\right)d\theta \\[2ex] | = \int_{0}^{\frac{\pi}{4}}\left(\sec^2(\theta) + 1\right)d\theta \\[2ex] | ||
&= \left[\tan\left({\frac{\pi}{4}}\right) + \frac{\pi}{4}\right] | |||
&= \tan({\theta}) + \theta \ \bigg|_{0}^{\frac{\pi}{4}}\\[2ex] | &= \tan({\theta}) + \theta \ \bigg|_{0}^{\frac{\pi}{4}}\\[2ex] | ||
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</math> | </math> | ||
&= \left[\tan\left({\frac{\pi}{4}}\right) + \frac{\pi}{4}\right] - left[\tan{0} + 0\right] \\[2ex] | &= \left[\tan\left({\frac{\pi}{4}}\right) + \frac{\pi}{4}\right] | ||
- left[\tan{0} + 0\right] \\[2ex] | |||
Revision as of 16:05, 21 September 2022
Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int _{0}^{\frac {\pi }{4}}\left({\frac {1+\cos ^{2}(\theta )}{\cos ^{2}(\theta )}}\right)d\theta &=\int _{0}^{\frac {\pi }{4}}\left({\frac {1}{\cos ^{2}(\theta )}}+{\frac {\cos ^{2}(\theta )}{\cos ^{2}(\theta )}}\right)d\theta =\int _{0}^{\frac {\pi }{4}}\left(\sec ^{2}(\theta )+1\right)d\theta \\[2ex]&=\left[\tan \left({\frac {\pi }{4}}\right)+{\frac {\pi }{4}}\right]&=\tan({\theta })+\theta \ {\bigg |}_{0}^{\frac {\pi }{4}}\\[2ex]&=1+{\frac {\pi }{4}}\end{aligned}}}
&= \left[\tan\left({\frac{\pi}{4}}\right) + \frac{\pi}{4}\right]
- left[\tan{0} + 0\right] \\[2ex]