Problem 22 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(B) \(\log |x| - f(x) + c\)

Step 1: Simplify the Integral

We are given two integrals. First, let's rewrite the second integral \( \int \frac{x^4}{x+x^5} \, dx \). Notice that \( \frac{x^4}{x+x^5} \) can be simplified as \( \frac{x^4}{x(1+x^4)} = \frac{x^3}{1+x^4} \). So our integral becomes \( \int \frac{x^3}{1+x^4} \, dx \).

Step 2: Use Substitution

Let us use substitution to simplify the integration process. Set \( u = 1 + x^4 \), which implies \( du = 4x^3 \, dx \). Therefore, \( x^3 \, dx = \frac{1}{4} \, du \). Substitute these into the integral to get: \( \int \frac{x^3}{1+x^4} \, dx = \frac{1}{4} \int \frac{1}{u} \, du \).

Step 3: Integrate Using Natural Logarithm

The integral \( \int \frac{1}{u} \, du \) is a basic natural logarithm integral. Integrating, we find \( \frac{1}{4} \int \frac{1}{u} \, du = \frac{1}{4} \log |u| + C \), where \( C \) is the constant of integration.

Step 4: Substitute Back in Terms of x

Replace \( u \) with \( 1 + x^4 \) in the integral result. So, \( \frac{1}{4} \log |1 + x^4| + C \). This is our antiderivative for the integral \( \int \frac{x^4}{x+x^5} \, dx \).

Step 5: Compare with Options and Solve

Now, based on the previously obtained information, \( \int \frac{1}{x + x^5} \, dx = f(x) + c \) could be related by substitution, it aligns with the option \( \log |x| - f(x) + c \) since \( f(x) = \log |1+x^4| \). Selecting option (B), \( \log |x| - f(x) + c \), is consistent.

Key Concepts

Substitution Method

The substitution method is a powerful integration technique that simplifies integrals by changing variables. It is especially useful when dealing with complex expressions. Here's how it works:

• Identify a part of the integral that can be substituted with a new variable. This usually involves the inner function in a composite function.
• Replace this part of the integral with the new variable. Substitute both the variable and its differential, so the integral is now in terms of the new variable.

In our example, we simplified the integral by substituting \( u = 1 + x^4 \), which led to \( du = 4x^3 \, dx \). By rearranging, we expressed \( x^3 \, dx \) as \( \frac{1}{4} \, du \) and substituted these into the original integral. This simplifies the integration process, making it more straightforward to handle.

Natural Logarithm Integration

Integrating functions involving the natural logarithm is common in calculus. When you encounter an integral of the form \( \int \frac{1}{u} \, du \), it results in a natural logarithm:

• The solution is \( \log |u| + C \), where \( C \) is the constant of integration.
• This occurs because the derivative of \( \log |u| \) is \( \frac{1}{u} \).

In the solution provided, after using substitution, the integral \( \int \frac{1}{u} \, du \) was evaluated as \( \frac{1}{4} \log |u| + C \). This simple result comes from recognizing the natural log integral form, which helps to determine antiderivatives easily. Remember, the absolute value in \( \log |u| \) ensures that the logarithm function domain includes negative inputs.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is crucial:

• An indefinite integral, represented as \( \int f(x) \, dx \), gives a function plus a constant \( C \): \( F(x) + C \).
• It represents a family of functions whose derivative equals the integrand.
• A definite integral, written \( \int_{a}^{b} f(x) \, dx \), computes a number representing the area under the function curve from \( a \) to \( b \).

In our exercise, the integral \( \int \frac{1}{x+x^5} \, dx \) is indefinite, leading to the expression \( f(x) + c \). The focus was on expressing one indefinite integral in terms of another. By evaluating and comparing, we determined that the expression relates to the given options, leading to the selection of the correct answer. This understanding highlights the subtle differences and similarities in approach when dealing with these types of integrals.