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$\int \tan^{2}(x) + 1 + \tan(x) \sec(x) \, dx$ =

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Gaussian integral

Full solution

Q.

\int \tan^{2}(x) + 1 + \tan(x) \sec(x) \, dx

=

Simplify using trigonometric identities: Simplify the integrand using trigonometric identities. $\newline$ We know that $\tan^2(x) + 1 = \sec^2(x)$ and that $\tan(x) \sec(x)$ is the derivative of $\sec(x)$ . So, the integral becomes: $\newline$ $\int(\tan^2(x) + 1 + \tan(x) \sec(x))\,dx = \int(\sec^2(x) + \tan(x) \sec(x))\,dx$
Split into two integrals: Split the integral into two separate integrals. $\newline$ \int(\sec^\(2(x) + \tan(x) \sec(x))\,dx = \int\sec^ $2$ (x)\,dx + \int\tan(x) \sec(x)\,dx
Integrate $\sec^2(x)$ : Integrate $\int \sec^2(x)\,dx$ . The integral of $\sec^2(x)$ with respect to $x$ is $\tan(x)$ , so we have: $\int \sec^2(x)\,dx = \tan(x)$
Integrate $\tan(x) \sec(x)$ : Integrate $\int \tan(x) \sec(x)\,dx$ . Since $\tan(x) \sec(x)$ is the derivative of $\sec(x)$ , the integral of $\tan(x) \sec(x)$ with respect to $x$ is $\sec(x)$ , so we have: $\int \tan(x) \sec(x)\,dx = \sec(x)$
Combine results: Combine the results from Step $3$ and Step $4$ . $\newline$ The integral of the original function is the sum of the integrals found in Step $3$ and Step $4$ , so we have: $\newline$ $\int(\tan^2(x) + 1 + \tan(x) \sec(x))\,dx = \tan(x) + \sec(x) + C$ , where $C$ is the constant of integration.

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