document.write( "Question 1207540: if f (x) + f ' (x) = 2 tan ^3(x) + tan ^2(x) + tan (x) + 1 , then f (x) = ... \n" ); document.write( "
Algebra.Com's Answer #847969 by CPhill(1987)\"\" \"About 
You can put this solution on YOUR website!
We are given a first-order linear differential equation:
\n" ); document.write( "```
\n" ); document.write( "f'(x) + f(x) = 2tan^3(x) + tan^2(x) + tan(x) + 1
\n" ); document.write( "```\r
\n" ); document.write( "\n" ); document.write( "This equation is in the form:
\n" ); document.write( "```
\n" ); document.write( "y' + P(x)y = Q(x)
\n" ); document.write( "```\r
\n" ); document.write( "\n" ); document.write( "where:
\n" ); document.write( "* y = f(x)
\n" ); document.write( "* P(x) = 1
\n" ); document.write( "* Q(x) = 2tan^3(x) + tan^2(x) + tan(x) + 1\r
\n" ); document.write( "\n" ); document.write( "**Solution using Integrating Factor:**\r
\n" ); document.write( "\n" ); document.write( "1. **Find the integrating factor:**
\n" ); document.write( " * Integrating factor, IF = e^(∫P(x)dx) = e^(∫1 dx) = e^x\r
\n" ); document.write( "\n" ); document.write( "2. **Multiply both sides of the equation by the integrating factor:**
\n" ); document.write( " * e^x * f'(x) + e^x * f(x) = e^x * (2tan^3(x) + tan^2(x) + tan(x) + 1)\r
\n" ); document.write( "\n" ); document.write( "3. **Recognize the left side as the derivative of a product:**
\n" ); document.write( " * d/dx (e^x * f(x)) = e^x * (2tan^3(x) + tan^2(x) + tan(x) + 1)\r
\n" ); document.write( "\n" ); document.write( "4. **Integrate both sides:**
\n" ); document.write( " * ∫d(e^x * f(x)) = ∫e^x * (2tan^3(x) + tan^2(x) + tan(x) + 1) dx\r
\n" ); document.write( "\n" ); document.write( "5. **Solve the integral on the right side:**
\n" ); document.write( " * This integral is quite complex and might require techniques like integration by parts or trigonometric substitutions. It's recommended to use a computer algebra system (like Wolfram Alpha) to evaluate this integral.\r
\n" ); document.write( "\n" ); document.write( "6. **Solve for f(x):**
\n" ); document.write( " * Once you've evaluated the integral, isolate f(x) by dividing both sides by e^x.\r
\n" ); document.write( "\n" ); document.write( "**Note:** The exact solution will involve the integral of the right-hand side, which might not have a simple closed-form expression. However, the general approach using the integrating factor is outlined above. \r
\n" ); document.write( "\n" ); document.write( "If you can provide more context or specific requirements, I might be able to assist further.
\n" ); document.write( " \n" ); document.write( "
\n" );