if `int (x^(3//2))/(x^(5//2) + x^4) dx = a "log" (x^b)/(1 + x^b) + C` then `(a,b)` is equal to

Options

A. `(3/2, 2/3)`
B. `(2/3 , 3/2)`
C. `(3/2,1)`
D. none

Answer

To solve the integral

\[ \int \frac{x^{3/2}}{x^{5/2} + x^4} \, dx, \]

we will simplify the integrand and then perform the integration step by step.

### Step 1: Simplify the integrand

First, we can factor out \(x^{5/2}\) from the denominator:

\[ x^{5/2} + x^4 = x^{5/2}(1 + x^{3/2}). \]

Thus, we can rewrite the integral as:

\[ \int \frac{x^{3/2}}{x^{5/2}(1 + x^{3/2})} \, dx = \int \frac{1}{x^{5/2}} \cdot \frac{x^{3/2}}{1 + x^{3/2}} \, dx. \]

This simplifies to:

\[ \int \frac{1}{x} \cdot \frac{1}{1 + x^{3/2}} \, dx. \]

### Step 2: Rewrite the integral

Now we have:

\[ \int \frac{1}{x(1 + x^{3/2})} \, dx. \]

### Step 3: Use partial fraction decomposition

We can express the integrand using partial fractions:

\[ \frac{1}{x(1 + x^{3/2})} = \frac{A}{x} + \frac{B}{1 + x^{3/2}}. \]

Multiplying through by the denominator \(x(1 + x^{3/2})\) gives:

\[ 1 = A(1 + x^{3/2}) + Bx. \]

### Step 4: Solve for coefficients A and B

To find \(A\) and \(B\), we can substitute convenient values for \(x\):

1. Let \(x = 0\):
\[ 1 = A(1 + 0) \implies A = 1. \]

2. Let \(x = 1\):
\[ 1 = 1(1 + 1) + B(1) \implies 1 = 2 + B \implies B = -1. \]

Thus, we have:

\[ \frac{1}{x(1 + x^{3/2})} = \frac{1}{x} - \frac{1}{1 + x^{3/2}}. \]

### Step 5: Integrate each term

Now we can integrate each term separately:

1. For \(\int \frac{1}{x} \, dx\):
\[ \int \frac{1}{x} \, dx = \log |x|. \]

2. For \(\int \frac{1}{1 + x^{3/2}} \, dx\), we can use substitution. Let \(u = 1 + x^{3/2}\), then \(du = \frac{3}{2} x^{1/2} \, dx\) or \(dx = \frac{2}{3} u^{-1/2} \, du\).

Thus, the integral becomes:

\[ -\int \frac{1}{u} \cdot \frac{2}{3} u^{-1/2} \, du = -\frac{2}{3} \log |u| = -\frac{2}{3} \log |1 + x^{3/2}|. \]

### Step 6: Combine results

Combining both integrals, we have:

\[ \int \frac{x^{3/2}}{x^{5/2} + x^4} \, dx = \log |x| - \frac{2}{3} \log |1 + x^{3/2}| + C. \]

### Step 7: Express in the required form

We can rewrite this as:

\[ \log |x| - \log |(1 + x^{3/2})^{2/3}| + C = \log \left(\frac{x}{(1 + x^{3/2})^{2/3}}\right) + C. \]

### Step 8: Compare with the given form

We need to express it in the form:

\[ a \log \left(\frac{x^b}{1 + x^b}\right) + C. \]

From our integration, we can see that:

- \(a = 2/3\)
- \(b = 3/2\)

Thus, the values of \(a\) and \(b\) are:

\[ (a, b) = \left(\frac{2}{3}, \frac{3}{2}\right). \]