Problem 22 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\int \frac{1}{x+x^{5}} d x=f(x)+c\), then \(\int \frac{x^{4}}{x+x^{5}} d x\) is equal to (A) \(\log |x|+f(x)+c\) (B) \(\log |x|-f(x)+c\) (C) \(x f(x)+c\) (D) none of these

Step-by-Step Solution

Answer

The answer is (D) none of these.

Step 1: Simplify the Given Integral

We need to evaluate the integral \(\int \frac{x^{4}}{x+x^{5}} dx\). First, notice that \(\frac{x^4}{x+x^5}\) can be rewritten as \(\frac{x^4}{x(1+x^4)}\), which simplifies to \(\frac{x^3}{1+x^4}\).

Step 2: Substitution Method

To simplify the integral, use the substitution \(u = x^4\). Then \(du = 4x^3 \, dx\), or \(x^3 \, dx = \frac{1}{4} \, du\). Substitute into the integral: \(\int \frac{x^3}{1+x^4} \, dx = \int \frac{1}{1+u} \cdot \frac{1}{4} \, du\).

Step 3: Evaluate the Substituted Integral

The integral becomes \(\frac{1}{4} \int \frac{1}{1+u} \, du\), which is a standard logarithmic integral. Integrating gives \(\frac{1}{4} \log |1+u| + C\).

Step 4: Re-substitute and Simplify the Result

Substitute \(u = x^4\) back into the expression to get \(\int \frac{x^4}{x+x^5} dx = \frac{1}{4} \log |1+x^4| + C\).

Step 5: Compare with the Given Options

Now compare our result \( \frac{1}{4} \log |1+x^4| + C\) with the given options (A), (B), (C), and (D). None of these options exactly match our result from evaluation.

Key Concepts

Substitution Method

In integral calculus, the substitution method is a powerful tool to simplify complex integrals. The main idea is to identify a part of the integral that can be transformed to make the integration process easier.

For instance, in the exercise, we encountered the integral \( \int \frac{x^4}{x+x^5} dx \). Originally challenging, this expression can be simplified via substitution. We let \( u = x^4 \), which helps transform the original function into a more integrable form.

When applying this method, remember to change the differential as well, meaning \( du = 4x^3 \, dx \), allowing us to express \( x^3 \, dx \) as \( \frac{1}{4} du \). Integrals with these new terms often allow for standard integration techniques to be used more effectively.

Logarithmic Integrals

Logarithmic integrals involve integrals that result in a logarithmic function. Recognizing such forms helps in direct evaluation of these integrals using standard formulas.

In our problem, after substitution, the integral \( \int \frac{x^3}{1+x^4} \, dx \) transforms into \( \int \frac{1}{1+u} \cdot \frac{1}{4} \, du \). This is easily identifiable as a standard logarithmic form. The integral of \( \frac{1}{1+u} \) is, of course, \( \log |1+u| \). By directly applying this knowledge, evaluating becomes straightforward.

Thus, utilizing the properties of logarithmic integrals, we achieve the integrated result \( \frac{1}{4} \log |1+u| + C \), streamlining the solving process significantly.

Simplifying Integrals

Simplifying integrals involves rewriting complex expressions into simpler forms that can be more easily integrated. This requires keen insight into algebraic manipulation and the properties of functions involved.

Initially, the integral \( \int \frac{x^4}{x+x^5} dx \) seems cumbersome. However, by factoring and rearranging, we rewrite it as \( \int \frac{x^3}{1+x^4} dx \). This simplification step is critical, setting up the problem for effective solving with substitution methods.

Successful simplification often makes use of identities and the understanding that each term in the integral can have relationships that, once uncovered, reduce complexity. This technique makes other methods, like substitution, executable and turns a seemingly challenging integral into one that aligns with readily solvable forms.