Question

Evaluate \[ \int \frac{1}{(x^4+1)^{5/4}}\,dx. \]

• A. \(-\dfrac{4}{(x^4+1)^{1/4}}\)
• B. \(\dfrac{1}{(x^4+1)^{1/4}}\)
• C. \(\dfrac{x}{(x^4+1)^{1/4}}\)
• D. \(-\dfrac{2}{(x^4+1)^{1/4}}\)

Hint:

In objective-type integration problems, differentiating the options is often faster than performing a full integration. The correct antiderivative differentiates exactly to the given integrand.

Answer 1

The Correct Option is C

Concept: For integrals of the form \[ \int \frac{1}{(x^4+1)^{5/4}}\,dx, \] it is convenient to verify the given options by differentiation.

Step 1: Differentiate Option (C). Let \[ F(x)=\frac{x}{(x^4+1)^{1/4}} = x(x^4+1)^{-1/4}. \] Using the product rule, \[ F'(x) = (x^4+1)^{-1/4} + x\left(-\frac14\right)(x^4+1)^{-5/4}(4x^3). \] \[ = (x^4+1)^{-1/4} - x^4(x^4+1)^{-5/4}. \] Taking \[ (x^4+1)^{-5/4} \] common, \[ F'(x) = (x^4+1)^{-5/4} \Big[(x^4+1)-x^4\Big]. \] \[ = (x^4+1)^{-5/4}. \] \[ = \frac{1}{(x^4+1)^{5/4}}. \]

Step 2: Compare with the integrand. Since \[ F'(x) = \frac{1}{(x^4+1)^{5/4}}, \] we have \[ \int \frac{1}{(x^4+1)^{5/4}}\,dx = \frac{x}{(x^4+1)^{1/4}} +C. \] Therefore, \[ \boxed{\int \frac{1}{(x^4+1)^{5/4}}\,dx = \frac{x}{(x^4+1)^{1/4}}+C} \] \[ \boxed{\text{Answer = (C)}} \]