The value of the integral `I=int(2x^(9)+x^(10))/((x^(2)+x^(3))^(3))dx` is equal to (where, C is the constant of integration)

Options

A. `(x^(4))/(2(1+x)^(2))+C`
B. `(x^(6))/(2(x+1)^(2))+C`
C. `(x^(4))/((x+1)^(2))+C`
D. `(x^(6))/(2(x+1)^(3))+C`

Answer

To solve the integral

\[ I = \int \frac{2x^9 + x^{10}}{(x^2 + x^3)^3} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Integral
First, we can factor out \(x^{12}\) from the numerator and denominator.

\[ I = \int \frac{x^{12} \left(2x^{-3} + x^{-2}\right)}{(x^2 + x^3)^3} \, dx. \]

### Step 2: Simplify the Denominator
Next, we simplify the denominator:

\[ (x^2 + x^3)^3 = x^6 (1 + x)^3. \]

Thus, the integral becomes:

\[ I = \int \frac{2x^{-3} + x^{-2}}{x^{-6}(1 + x)^3} \, dx = \int \frac{2x^3 + x^4}{(1 + x)^3} \, dx. \]

### Step 3: Rewrite the Integral
Now we can rewrite the integral:

\[ I = \int \frac{2x^3 + x^4}{(1 + x)^3} \, dx. \]

### Step 4: Substitute
Let \(u = 1 + x\), then \(du = dx\) and \(x = u - 1\). Therefore, we have:

\[ I = \int \frac{2(u - 1)^3 + (u - 1)^4}{u^3} \, du. \]

### Step 5: Expand the Numerator
Now we expand the numerator:

\[ 2(u - 1)^3 = 2(u^3 - 3u^2 + 3u - 1) = 2u^3 - 6u^2 + 6u - 2, \]
\[ (u - 1)^4 = u^4 - 4u^3 + 6u^2 - 4u + 1. \]

Combining these gives:

\[ 2(u - 1)^3 + (u - 1)^4 = (2u^3 - 6u^2 + 6u - 2) + (u^4 - 4u^3 + 6u^2 - 4u + 1). \]

This simplifies to:

\[ u^4 - 2u^3 + 0u^2 + 2u - 1. \]

### Step 6: Rewrite the Integral
Thus, we have:

\[ I = \int \frac{u^4 - 2u^3 + 2u - 1}{u^3} \, du = \int (u - 2 + \frac{2}{u^2} - \frac{1}{u^3}) \, du. \]

### Step 7: Integrate Term by Term
Now we can integrate term by term:

\[ I = \int u \, du - 2\int 1 \, du + 2\int u^{-2} \, du - \int u^{-3} \, du. \]

This results in:

\[ I = \frac{u^2}{2} - 2u - \frac{2}{u} + \frac{1}{2u^2} + C. \]

### Step 8: Substitute Back
Now we substitute back \(u = 1 + x\):

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

### Final Step: Simplify the Expression
After simplifying, we can express the integral in a more compact form.

The final answer is:

\[ I = \frac{x^4}{2(1 + x)^2} + C. \]