Question

Evaluate \[ \int \frac{x^7-1} {x^8\sqrt{\,1-2x^7+7x^{14}\,}} \,dx. \]

• A. \[ \frac{\sqrt{1-2x^7+7x^{14}}}{7x^7}+c \]
• B. \[ \log\!\left(\sqrt{1-2x^7+7x^{14}}\right)+c \]
• C. \[ \frac1{x^8}-\frac1{x^{15}}+c \]
• D. \[ \sqrt{\frac1{x^8}-\frac2{x^7}+\frac7{x^{14}}}+c \]

Hint:

Whenever the integrand is of the form \[ \boxed{ \frac{f'(x)} {\sqrt{f(x)}}, } \] or resembles the derivative of a quotient involving \[ \sqrt{f(x)}, \] try differentiating the given options to identify the correct antiderivative.

Answer 1

The Correct Option is A

Step 1: Choose a suitable substitution. Let \[ u=\frac{\sqrt{1-2x^7+7x^{14}}}{7x^7}. \] Differentiate \(u\). Using the quotient rule together with the chain rule, \[ \frac{du}{dx} = \frac{x^7-1} {x^8\sqrt{1-2x^7+7x^{14}}}. \]

Step 2: Integrate. Hence, \[ \int \frac{x^7-1} {x^8\sqrt{1-2x^7+7x^{14}}} \,dx = \int du = u+c. \] Substituting back, \[ \boxed{ \int \frac{x^7-1} {x^8\sqrt{1-2x^7+7x^{14}}} \,dx = \frac{\sqrt{1-2x^7+7x^{14}}}{7x^7}+c. } \] Thus, \[ \boxed{(A)} \] is the correct answer.