If `int1/(x+x^5)dx=f(x)+c ,t h e ne v a l u a t eint(x^4)/(x+x^5)dxdot`

Explanation

To solve the given problem step by step, we start with the integral:

\[ \int \frac{1}{x + x^5} \, dx = f(x) + c \]

We need to evaluate the integral:

\[ \int \frac{x^4}{x + x^5} \, dx \]

### Step 1: Simplify the integrand

We can rewrite the integrand \(\frac{x^4}{x + x^5}\) by factoring out \(x\) from the denominator:

\[ \frac{x^4}{x + x^5} = \frac{x^4}{x(1 + x^4)} = \frac{x^3}{1 + x^4} \]

### Step 2: Rewrite the integral

Now, we can express the integral as:

\[ \int \frac{x^3}{1 + x^4} \, dx \]

### Step 3: Use substitution

Let’s use the substitution \(u = 1 + x^4\). Then, we differentiate:

\[ du = 4x^3 \, dx \quad \Rightarrow \quad dx = \frac{du}{4x^3} \]

Now, substituting \(u\) into the integral:

\[ \int \frac{x^3}{u} \cdot \frac{du}{4x^3} = \frac{1}{4} \int \frac{1}{u} \, du \]

### Step 4: Integrate

The integral of \(\frac{1}{u}\) is:

\[ \frac{1}{4} \ln |u| + C = \frac{1}{4} \ln |1 + x^4| + C \]

### Step 5: Combine with known integral

We know from the problem statement that:

\[ \int \frac{1}{x + x^5} \, dx = f(x) + c \]

Thus, we can express our integral as:

\[ \int \frac{x^4}{x + x^5} \, dx = \frac{1}{4} \ln |1 + x^4| + C - (f(x) + c) \]

### Final Result

Therefore, the evaluated integral is:

\[ \int \frac{x^4}{x + x^5} \, dx = \frac{1}{4} \ln |1 + x^4| - f(x) + C' \]

where \(C' = C - c\) is a new constant of integration.

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