integral of sec^2(x) Archives - Epsilonify Best solutions on internet! Wed, 13 Sep 2023 20:36:42 +0000 en-US hourly 1 https://wordpress.org/?v=6.4.5 https://www.epsilonify.com/wp-content/uploads/2022/09/cropped-E-M7-32x32.png integral of sec^2(x) Archives - Epsilonify 32 32 What is the integral of sec^2(x)? https://www.epsilonify.com/mathematics/calculus/what-is-the-integral-of-sec-square-x/ https://www.epsilonify.com/mathematics/calculus/what-is-the-integral-of-sec-square-x/#respond Sat, 14 Jan 2023 13:00:10 +0000 https://www.epsilonify.com/?p=1882 The integral of is . Proof. In this case, it is easy to show what the integral is if you have seen derivatives. Notice that we have seen here that Taking the antiderivative of , we do directly see that: What about the integral of ? We do know that the derivative of is by […]

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]]> \sec^2(ax) is \frac{1}{a}\tan(ax) + C.

Proof. In this case, it is easy to show what the integral \sec^2(ax) is if you have seen derivatives. Notice that we have seen here that 
\begin{align*}
\frac{d}{dx} \tan(x) = \sec^2(x). 
\end{align*}
Taking the antiderivative of \sec^2(x), we do directly see that: 
\begin{align*}
\int \sec^2(x)dx = \tan(x) + C'.
\end{align*}
What about the integral of \sec^2(ax)? We do know that the derivative of \frac{1}{a}\tan(ax) is \sec^2(ax) by using the chain rule. Therefore, taking antiderivative of \sec^2(ax), we get: 
\begin{align*}
\int \sec^2(ax)dx = \frac{1}{a}\tan(ax) + C.
\end{align*}
So, the integral of \sec^2(ax) is \frac{1}{a}\tan(ax) + C.

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