`intx^(-2//3)(1+x^(1//2))^(-5//3)` dx is equal to

Options

A. `3(1+x^(-1//2))^(-1//3)+C`
B. `3(1+x^(-1//2))^(-2//3)+C`
C. `3(1+x^(1//2))^(-2//3)+C`
D. none of these

Answer

To solve the integral \( I = \int x^{-\frac{2}{3}} (1 + x^{\frac{1}{2}})^{-\frac{5}{3}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the given integral:
\[ I = \int x^{-\frac{2}{3}} (1 + x^{\frac{1}{2}})^{-\frac{5}{3}} \, dx \]

### Step 2: Factor Out \( x^{\frac{1}{2}} \)
Notice that we can factor out \( x^{\frac{1}{2}} \) from the term \( (1 + x^{\frac{1}{2}})^{-\frac{5}{3}} \):
\[ I = \int x^{-\frac{2}{3}} \left( x^{\frac{1}{2}} \left( \frac{1}{x^{\frac{1}{2}}} + 1 \right) \right)^{-\frac{5}{3}} \, dx \]
This simplifies to:
\[ I = \int x^{-\frac{2}{3}} x^{-\frac{5}{6}} \left( \frac{1}{x^{\frac{1}{2}}} + 1 \right)^{-\frac{5}{3}} \, dx \]

### Step 3: Combine Exponents
Now, we combine the exponents:
\[ I = \int x^{-\frac{2}{3} - \frac{5}{6}} \left( \frac{1}{x^{\frac{1}{2}}} + 1 \right)^{-\frac{5}{3}} \, dx \]
Finding a common denominator for the exponents:
\[ -\frac{2}{3} = -\frac{4}{6} \quad \text{and} \quad -\frac{5}{6} = -\frac{5}{6} \]
Thus:
\[ -\frac{4}{6} - \frac{5}{6} = -\frac{9}{6} = -\frac{3}{2} \]
So we have:
\[ I = \int x^{-\frac{3}{2}} \left( \frac{1}{x^{\frac{1}{2}}} + 1 \right)^{-\frac{5}{3}} \, dx \]

### Step 4: Substitute
Let \( t = x^{-\frac{1}{2}} + 1 \). Then, differentiating gives:
\[ dt = -\frac{1}{2} x^{-\frac{3}{2}} \, dx \quad \Rightarrow \quad dx = -2 x^{\frac{3}{2}} \, dt \]
Substituting \( x^{-\frac{3}{2}} \) gives us:
\[ x^{-\frac{3}{2}} = -2 dt \]
Thus, we can rewrite the integral:
\[ I = -2 \int t^{-\frac{5}{3}} \, dt \]

### Step 5: Integrate
Now we integrate:
\[ I = -2 \left( \frac{t^{-\frac{5}{3} + 1}}{-\frac{5}{3} + 1} \right) + C \]
This simplifies to:
\[ I = -2 \left( \frac{t^{-\frac{2}{3}}}{\frac{-2}{3}} \right) + C = 3 t^{-\frac{2}{3}} + C \]

### Step 6: Substitute Back
Now substituting back for \( t \):
\[ I = 3 \left( x^{-\frac{1}{2}} + 1 \right)^{-\frac{2}{3}} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ I = 3 (1 + x^{-\frac{1}{2}})^{-\frac{2}{3}} + C \]