If `I=int(5x^(8)+7x^(6))/((x^(2)+1+2x^(7))^(2))dx` then l is equal to

Options

A. `(x^(7))/(2x^(7)+x^(2)+1)+C`
B. `(x^(5))/(x^(2)+1+2x^(7))+C`
C. `(-1)/(2x^(7)+x^(2)+1)+C`
D. `(p(x))/(q(x))," degp "(x)="deg q"(x)=7`

Answer

To solve the integral \( I = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} \, dx \), we will follow these steps:

### Step 1: Simplify the Integral
We start with the integral:

\[ I = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} \, dx \]

### Step 2: Identify the Denominator
Notice that the denominator is a polynomial raised to the power of 2. We can denote:

\[ t = x^2 + 1 + 2x^7 \]

### Step 3: Differentiate \( t \)
Next, we differentiate \( t \) with respect to \( x \):

\[ \frac{dt}{dx} = 2x + 14x^6 \]

This means:

\[ dt = (2x + 14x^6) \, dx \]

### Step 4: Express \( dx \) in Terms of \( dt \)
We can rearrange this to express \( dx \):

\[ dx = \frac{dt}{2x + 14x^6} \]

### Step 5: Substitute \( t \) and \( dx \) into the Integral
Now we substitute \( t \) and \( dx \) back into the integral:

\[ I = \int \frac{5x^8 + 7x^6}{t^2} \cdot \frac{dt}{2x + 14x^6} \]

### Step 6: Simplify the Integral
We can simplify the expression inside the integral. The numerator can be factored or simplified based on \( t \):

\[ 5x^8 + 7x^6 = x^6(5x^2 + 7) \]

### Step 7: Substitute Back to Original Variables
After performing the integration, we will substitute back \( t \) to express the result in terms of \( x \):

\[ I = -\frac{1}{t} + C = -\frac{1}{x^2 + 1 + 2x^7} + C \]

### Final Result
Thus, the final result for the integral is:

\[ I = -\frac{1}{x^2 + 1 + 2x^7} + C \]