Question:
If
\[
\int \frac{x^8+4}{x^4-2x^2+2}\,dx=Ax^5+Bx^3+Cx+K,
\]
then \(5A+3B+C=\)
If
\[
\int \frac{x^8+4}{x^4-2x^2+2}\,dx=Ax^5+Bx^3+Cx+K,
\]
then \(5A+3B+C=\)
Show Hint
Before integrating a rational algebraic expression, first check whether the numerator is exactly divisible by the denominator. It can make the integration very simple.
Updated On: Jun 26, 2026
- \(7\)
- \(5\)
- \(3\)
- \(1\)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Simplify the integrand.
We have \[ \frac{x^8+4}{x^4-2x^2+2}. \] By polynomial division, \[ x^8+4=(x^4-2x^2+2)(x^4+2x^2+2). \] Therefore, \[ \frac{x^8+4}{x^4-2x^2+2}=x^4+2x^2+2. \]
Step 2: Integrate the simplified expression.
Now, \[ \int \frac{x^8+4}{x^4-2x^2+2}\,dx = \int (x^4+2x^2+2)\,dx. \] So, \[ \int (x^4+2x^2+2)\,dx = \frac{x^5}{5}+\frac{2x^3}{3}+2x+K. \]
Step 3: Compare with the given expression.
Given, \[ \int \frac{x^8+4}{x^4-2x^2+2}\,dx=Ax^5+Bx^3+Cx+K. \] Comparing, \[ A=\frac{1}{5},\quad B=\frac{2}{3},\quad C=2. \]
Step 4: Find \(5A+3B+C\).
Now, \[ 5A+3B+C = 5\left(\frac{1}{5}\right)+3\left(\frac{2}{3}\right)+2. \] \[ =1+2+2. \] \[ =5. \]
Step 5: Final conclusion.
Therefore, \[ \boxed{5} \]
We have \[ \frac{x^8+4}{x^4-2x^2+2}. \] By polynomial division, \[ x^8+4=(x^4-2x^2+2)(x^4+2x^2+2). \] Therefore, \[ \frac{x^8+4}{x^4-2x^2+2}=x^4+2x^2+2. \]
Step 2: Integrate the simplified expression.
Now, \[ \int \frac{x^8+4}{x^4-2x^2+2}\,dx = \int (x^4+2x^2+2)\,dx. \] So, \[ \int (x^4+2x^2+2)\,dx = \frac{x^5}{5}+\frac{2x^3}{3}+2x+K. \]
Step 3: Compare with the given expression.
Given, \[ \int \frac{x^8+4}{x^4-2x^2+2}\,dx=Ax^5+Bx^3+Cx+K. \] Comparing, \[ A=\frac{1}{5},\quad B=\frac{2}{3},\quad C=2. \]
Step 4: Find \(5A+3B+C\).
Now, \[ 5A+3B+C = 5\left(\frac{1}{5}\right)+3\left(\frac{2}{3}\right)+2. \] \[ =1+2+2. \] \[ =5. \]
Step 5: Final conclusion.
Therefore, \[ \boxed{5} \]
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