5.4 Indefinite Integrals and the Net Change Theorem/17: Difference between revisions

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17)<math>\int{}{}1+tan^{2}x*dx = \frac{d}{dx}(tan(x))</math>
<math>
\int(1+\tan^2{\alpha})\,d\alpha = \int\sec^2\alpha \,d\alpha = \tan\alpha + C
</math>


<math>\frac{d}{dx}(tan(x))=\frac{d}{dx}\left(\frac{sin(x)}{cos(x)}\right)=\frac{cos^{2}x-(-(sin(x))(sin(x))}{cos^{2}(x)}</math>


Note: <math>1+\tan^2{\alpha} = \sec^2\alpha</math>


<math>=\frac{cos^{2}x+sin^2(x))}{cos^{2}(x)}=\frac{cos^{2}x}{cos^{2}(x)}+\frac{sin^{2}x}{cos^{2}(x)}</math>


Or,


<math>=1+tan^2(x)</math>


<math>


[[5.3 The Fundamental Theorem of Calculus/1|1]]
\int(1+\tan^2{\alpha})\,d\alpha = \int\left(1+\frac{sin^2\alpha}{cos^2\alpha}\right)d\alpha = \int\left(\frac{cos^2\alpha+sin^2\alpha}{cos^2\alpha}\right)d\alpha  = \int\frac{1}{cos^2\alpha}\,d\alpha =
[[5.3 The Fundamental Theorem of Calculus/3|3]]
 
[[5.3 The Fundamental Theorem of Calculus/5|5]]
\int\sec^2\alpha \,d\alpha = \tan{\alpha}+C
[[5.3 The Fundamental Theorem of Calculus/7|7]]
</math>
[[5.3 The Fundamental Theorem of Calculus/8|8]]
 
[[5.3 The Fundamental Theorem of Calculus/9|9]]
 
[[5.3 The Fundamental Theorem of Calculus/10|10]]
Note: <math>\cos^2\alpha+sin^2\alpha=1</math>
[[5.3 The Fundamental Theorem of Calculus/11|11]]
[[5.3 The Fundamental Theorem of Calculus/13|13]]
[[5.3 The Fundamental Theorem of Calculus/15|15]]
[[5.3 The Fundamental Theorem of Calculus/17|17]]
[[5.3 The Fundamental Theorem of Calculus/19|19]]
[[5.3 The Fundamental Theorem of Calculus/20|20]]
[[5.3 The Fundamental Theorem of Calculus/21|21]]
[[5.3 The Fundamental Theorem of Calculus/23|23]]
[[5.3 The Fundamental Theorem of Calculus/25|25]]
[[5.3 The Fundamental Theorem of Calculus/27|27]]
[[5.3 The Fundamental Theorem of Calculus/28|28]]
[[5.3 The Fundamental Theorem of Calculus/29|29]]
[[5.3 The Fundamental Theorem of Calculus/31|31]]
[[5.3 The Fundamental Theorem of Calculus/33|33]]
[[5.3 The Fundamental Theorem of Calculus/35|35]]
[[5.3 The Fundamental Theorem of Calculus/37|37]]
[[5.3 The Fundamental Theorem of Calculus/39|39]]
[[5.3 The Fundamental Theorem of Calculus/41|41]]
[[5.3 The Fundamental Theorem of Calculus/53|53]]

Latest revision as of 19:39, 21 September 2022


Note:


Or,



Note:

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