Question

evaluate the integral (3x^-7 +x^-1) dx​

Answer 1

Answer:

$\mathrm{\dfrac{-1}{2x^6} + log|x|+c}$

Step-by-step explanation:

Given :

$\mathrm {\displaystyle \int (3x^{-7} + x^{-1}) \, dx }$

To find :

Evaluate the given integral

Solution :

$\longmapsto \mathrm {\displaystyle \int (3x^{-7} + x^{-1}) \, dx }$

$\implies \mathrm {\displaystyle \int (3x^{-7} + \dfrac{1}{x}) \, dx }$

We know that,

$\boxed{\begin{array}{cc} \mathrm {\displaystyle \int x^n \, dx = \dfrac{x^{n+1}}{n+1} + c \ \ \ (n \neq -1)}\\\\\mathrm {\displaystyle \int \dfrac{1}{x} \, dx = log|x| + c} \end{array}}$

So, applying these

$\implies \mathrm {\displaystyle \int 3x^{-7} \, dx + \displaystyle \int \dfrac{1}{x} \, dx }$

$\implies \mathrm{\dfrac{3x^{-6}}{-6} + log|x| + c}$

$\implies \mathrm{\dfrac{-\not3x^{-6}}{\not6^2} + log|x| + c}$

$\implies \mathrm{\dfrac{-1}{2x^6} + log|x|+c}$

$\therefore \underline{\boxed{\mathbf {\displaystyle \int (3x^{-7} + x^{-1}) \, dx}= \mathbf{\dfrac{-1}{2x^6} + log|x|+c} }}$

❖ Extra information :

⟡ Some basic integrals :

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\\\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx+c \\\\ \sf x^n \ (n \neq -1)& \sf \dfrac{x^{n+1}}{n+1} + c \\\\ \sf \dfrac{1}{x} & \sf logx+ c\\\\ \sf {e}^{x} & \sf {e}^{x}+c\\\\ \sf sinx & \sf - \: cosx+ c \\\\ \sf cosx & \sf \: sinx + c\\\\ \sf {sec}^{2} x & \sf tanx + c\\\\ \sf {cosec}^{2}x & \sf - cotx+ c \\\\ \sf secx \: tanx & \sf secx + c\\\\ \sf cosecx \: cotx& \sf -\: cosecx + c\end{array}} \\ \end{gathered}$

Hope it helps!!