Question

Evaluate: \[ \int \frac{dx}{(x^5+1)^{6/5}} \]

• A. \(\dfrac{1}{\sqrt[5]{x^5+1}}+c\)
• B. \(\dfrac{\sqrt[5]{x^5+1}}{x}+c\)
• C. \(\dfrac{x}{\sqrt[5]{x^5+1}}+c\)
• D. \(\sqrt[5]{x^5+1}+c\)

Hint:

Whenever an integrand contains expressions like \[ (x^n+a)^{-\frac{n+1}{n}} \] try differentiating \[ \frac{x}{(x^n+a)^{1/n}} \] because product rule often simplifies beautifully.

Answer 1

The Correct Option is C

Concept: Integrals involving expressions of the form: \[ (x^n+a)^{m} \] often simplify through substitution or by identifying the derivative of a cleverly chosen expression. Here, the integrand strongly suggests differentiating: \[ \frac{x}{(x^5+1)^{1/5}} \] because the power \(-\frac65\) appears naturally after differentiation.

Step 1: Assume a likely antiderivative. Consider: \[ F(x)=\frac{x}{(x^5+1)^{1/5}} \] Rewrite as: \[ F(x)=x(x^5+1)^{-1/5} \] Differentiate using the product rule.

Step 2: Differentiate carefully. \[ F'(x) = (x^5+1)^{-1/5} + x\left(-\frac15\right)(x^5+1)^{-6/5}(5x^4) \] Simplifying, \[ F'(x) = (x^5+1)^{-1/5} - x^5(x^5+1)^{-6/5} \] Taking common factor: \[ F'(x) = (x^5+1)^{-6/5}\Big[(x^5+1)-x^5\Big] \] \[ F'(x) = (x^5+1)^{-6/5} \] Therefore, \[ F'(x)=\frac1{(x^5+1)^{6/5}} \] which matches the integrand exactly.

Step 3: Write the integral directly. Hence, \[ \int \frac{dx}{(x^5+1)^{6/5}} = \frac{x}{(x^5+1)^{1/5}}+c \] or equivalently, \[ \boxed{ \frac{x}{\sqrt[5]{x^5+1}}+c } \]