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

Step 1: Guess the antiderivative.
The power $-\dfrac65$ in the integrand suggests the answer looks like $\dfrac{x}{(x^5+1)^{1/5}}$, because differentiating it should drop the power to $-\dfrac65$.
Step 2: Set up the trial function.
Let $F(x)=x(x^5+1)^{-1/5}$. We will differentiate it using the product rule.
Step 3: Differentiate term by term.
\[ F'(x)=(x^5+1)^{-1/5}+x\left(-\frac15\right)(x^5+1)^{-6/5}(5x^4). \]
Step 4: Simplify the second term.
$x\cdot\left(-\dfrac15\right)\cdot5x^4=-x^5$, so \[ F'(x)=(x^5+1)^{-1/5}-x^5(x^5+1)^{-6/5}. \]
Step 5: Take a common factor.
Factor out $(x^5+1)^{-6/5}$: \[ F'(x)=(x^5+1)^{-6/5}\big[(x^5+1)-x^5\big]=(x^5+1)^{-6/5}. \] This is exactly the integrand.
Step 6: Write the integral.
\[ \int\frac{dx}{(x^5+1)^{6/5}}=\frac{x}{\sqrt[5]{x^5+1}}+c. \] \[ \boxed{\dfrac{x}{\sqrt[5]{x^5+1}}+c} \]