`int x^(13/2) (1+x^(5/2))^(1/2) dx = P(1+x^(5/2))^(7/2)+Q(1+x^(5/2))^(5/2) +R(1+x^(5/2))^(3/2)+C` then P, Q ,R are

Options

A. `A=-(4)/(35),B=-(8)/(25),C=(4)/(15)`
B. `A=-(4)/(35),B=(8)/(25),C=(4)/(15)`
C. `A=(+4)/(35),B=(-8)/(25),C=(+4)/(15)`
D. None of these

Answer

To solve the integral

\[ I = \int x^{\frac{13}{2}} \sqrt{1 + x^{\frac{5}{2}}} \, dx, \]

we will use a substitution method. Let's follow the steps:

### Step 1: Substitution
Let
\[ T = 1 + x^{\frac{5}{2}}. \]
Then, differentiate both sides:
\[ \frac{dT}{dx} = \frac{5}{2} x^{\frac{3}{2}} \implies dT = \frac{5}{2} x^{\frac{3}{2}} \, dx. \]
From this, we can express \(dx\) in terms of \(dT\):
\[ dx = \frac{2}{5} \frac{dT}{x^{\frac{3}{2}}}. \]

### Step 2: Express \(x\) in terms of \(T\)
From our substitution \(T = 1 + x^{\frac{5}{2}}\), we can express \(x^{\frac{5}{2}}\) as:
\[ x^{\frac{5}{2}} = T - 1 \implies x = (T - 1)^{\frac{2}{5}}. \]

### Step 3: Substitute \(dx\) and \(x\) in the integral
Now substituting \(dx\) and \(x^{\frac{13}{2}}\) in terms of \(T\):
\[ x^{\frac{13}{2}} = \left((T - 1)^{\frac{2}{5}}\right)^{\frac{13}{2}} = (T - 1)^{\frac{13}{5}}. \]
Now substitute into the integral:
\[ I = \int (T - 1)^{\frac{13}{5}} \sqrt{T} \cdot \frac{2}{5} \frac{dT}{(T - 1)^{\frac{3}{5}}}. \]
This simplifies to:
\[ I = \frac{2}{5} \int (T - 1)^{\frac{13}{5} - \frac{3}{5}} T^{\frac{1}{2}} \, dT = \frac{2}{5} \int (T - 1)^{\frac{10}{5}} T^{\frac{1}{2}} \, dT = \frac{2}{5} \int (T - 1)^2 T^{\frac{1}{2}} \, dT. \]

### Step 4: Expand and integrate
Now we expand \((T - 1)^2\):
\[ (T - 1)^2 = T^2 - 2T + 1. \]
Thus,
\[ I = \frac{2}{5} \int (T^2 - 2T + 1) T^{\frac{1}{2}} \, dT = \frac{2}{5} \int (T^{\frac{5}{2}} - 2T^{\frac{3}{2}} + T^{\frac{1}{2}}) \, dT. \]
Now we can integrate term by term:
\[ I = \frac{2}{5} \left( \frac{T^{\frac{7}{2}}}{\frac{7}{2}} - 2 \cdot \frac{T^{\frac{5}{2}}}{\frac{5}{2}} + \frac{T^{\frac{3}{2}}}{\frac{3}{2}} \right) + C. \]
This simplifies to:
\[ I = \frac{2}{5} \left( \frac{2}{7} T^{\frac{7}{2}} - \frac{8}{5} T^{\frac{5}{2}} + \frac{2}{3} T^{\frac{3}{2}} \right) + C. \]

### Step 5: Substitute back for \(T\)
Now, substitute back \(T = 1 + x^{\frac{5}{2}}\):
\[ I = \frac{4}{35} (1 + x^{\frac{5}{2}})^{\frac{7}{2}} - \frac{8}{25} (1 + x^{\frac{5}{2}})^{\frac{5}{2}} + \frac{4}{15} (1 + x^{\frac{5}{2}})^{\frac{3}{2}} + C. \]

### Step 6: Identify coefficients
From the expression, we can identify:
- \(P = \frac{4}{35}\)
- \(Q = -\frac{8}{25}\)
- \(R = \frac{4}{15}\)

### Final Answer
Thus, the values of \(P\), \(Q\), and \(R\) are:
\[ P = \frac{4}{35}, \quad Q = -\frac{8}{25}, \quad R = \frac{4}{15}. \]