Question

Evaluate the integral by making the given substitution. (use c for the constant of integration.) sec2(1/x9) x10 dx, u = 1/x9

Answer 1

Answer:

$\frac{1}{-9}tan \frac{1}{x^9} + c$

Step-by-step explanation:

The question requires addition information. Below is how the question should be rightly stated.

        $\int\limits {\frac{sec^{2}(\frac{1}{x^9})}{ x^{10} } } \, dx$   ------(1)

Using substitution method,

We are given $u =\frac{1}{x^{9}}$                   -------(2)

                      ⇒ $u =x^{-9}$

Differentiating u with respect to x,

                      $\frac{du}{dx} = -9x^{-9-1}$

                      $\frac{du}{dx} = -9x^{-10}$

Making dx the subject of the equation

                      $dx =\frac{du}{-9x^{-10}}$

                      $dx =\frac{x^{10}du}{-9}$                         ---------(3)

Substituting the values of u from equation (2) and dx from equation (3) into equation (1)

                      $\int\limits {\frac{sec^{2}u}{x^{10}}} . \frac{x^{10}}{-9} \, du$

                      $\int\limits {\frac{sec^{2}u}{-9}} \, du$

                      $\frac{1}{-9}\int\limits {sec^{2}u \, du$

                      $\frac{1}{-9}tanu + c$

substituting the value of u

                      $\frac{1}{-9}tan \frac{1}{x^9} + c$