Question

7D%7Bc%7Cc%7D%20%5Cbf%20f%28x%29%20%26%20%5Cbf%20%5Cdisplaystyle%20%5Cint%20%5Crm%20%5C%3Af%28x%29%20%5C%3A%20dx%5C%5C%20%5C%5C%20%5Cfrac%7B%5Cqquad%20%5Cqquad%7D%7B%7D%20%26%20%5Cfrac%7B%5Cqquad%20%5Cqquad%7D%7B%7D%20%5C%5C%20%5Csf%20k%20%26%20%5Csf%20kx%20%2B%20c%20%5C%5C%20%5C%5C%20%5Csf%20sinx%20%26%20%5Csf%20-%20%5C%3A%20cosx%2B%20c%20%5C%5C%20%5C%5C%20%5Csf%20cosx%20%26%20%5Csf%20%5C%3A%20sinx%20%2B%20c%5C%5C%20%5C%5C%20%5Csf%20%7Bsec%7D%5E%7B2%7D%20x%20%26%20%5Csf%20tanx%20%2B%20c%5C%5C%20%5C%5C%20%5Csf%20%7Bcosec%7D%5E%7B2%7Dx%20%26%20%5Csf%20-%20cotx%2B%20c%20%5C%5C%20%5C%5C%20%5Csf%20secx%20%5C%3A%20tanx%20%26%20%5Csf%20secx%20%2B%20c%5C%5C%20%5C%5C%20%5Csf%20cosecx%20%5C%3A%20cotx%26%20%5Csf%20-%20%5C%3A%20cosecx%20%2B%20c%5C%5C%20%5C%5C%20%5Csf%20tanx%20%26%20%5Csf%20logsecx%20%2B%20c%5C%5C%20%5C%5C%20%5Csf%20%5Cdfrac%7B1%7D%7Bx%7D%20%26%20%5Csf%20logx%2B%20c%5C%5C%20%5C%5C%20%5Csf%20%7Be%7D%5E%7Bx%7D%20%26%20%5Csf%20%7Be%7D%5E%7Bx%7D%20%2B%20c%5Cend%7Barray%7D%7D%20%5C%5C%20%5Cend%7Bgathered%7D%5Cend%7Bgathered%7D%5Cend%7Bgathered%7D" id="TexFormula1" title="\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}\end{gathered}" alt="\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}\end{gathered}" align="absmiddle" class="latex-formula">
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Answer 1

$\int \frac{dx}{x} = log(x) + c \\ \int \frac{f \prime(x)}{f(x)} = log \mid \: f(x) \mid + c \\ \int \: tan(x)dx = \int \: \frac{sin(x)}{cos(x)} dx \\ = - log|cos(x)| + c \\ = log |sec(x)| + c \\ the \: same \: rule \: goes \: for \: cot...etc.\\\int {sec}^{2}(x)dx=|tan(x)|+c\\let u=tanx\\ \frac{du}{dx}={sec}^{2}(x)\rightarrow {du}={sec}^{2}(x)dx\\ \int du=|u|+c\\ \therefore tan|x|+c$

Step-by-step explanation:

Am not sure what your question is? But if you are asking about a proof, then you may use Taylor series to prove these integrals...