The problem to be solved is: ∫e^8x * sin(x) dx
The homework equations that need to be used are:
∫e^ax * sin(bx) dx = (a^2 + b^2)^-1 * e^ax * (a * sin(bx) - b * cos(bx)) + C
∫e^ax * cos(bx) dx = (a^2 + b^2)^-1 * e^ax * (a * cos(bx) + b * sin(bx)) + C
The general strategy for solving integration questions is to first identify the appropriate technique or formula to use, then apply any necessary algebraic manipulations or substitutions, and finally evaluate the integral using basic integration rules.
To solve the given integration problem, we first use the formula ∫e^ax * sin(bx) dx = (a^2 + b^2)^-1 * e^ax * (a * sin(bx) - b * cos(bx)) + C with a = 8 and b = 1. This gives us:
∫e^8x * sin(x) dx = (8^2 + 1^2)^-1 * e^8x * (8 * sin(x) - 1 * cos(x)) + C
= 1/65 * e^8x * (8 * sin(x) - cos(x)) + C
Next, we apply the formula ∫e^ax * cos(bx) dx = (a^2 + b^2)^-1 * e^ax * (a * cos(bx) + b * sin(bx)) + C with a = 8 and b = 1. This gives us:
∫e^8x * cos(x) dx = (8^2 + 1^2)^-1 * e^8x * (8 * cos(x) + 1 * sin(x)) + C
= 1/65 * e^8x * (8 * cos(x) + sin(x)) + C
Finally, we combine these two results to get the final answer:
∫e^8x * sin(x) dx = 1/65 * e^8x * (8 * sin(x) - cos(x) + 8 * cos(x) + sin(x)) + C
= 1/65 * e^8x * (9 * sin(x) + 7 * cos(x)) + C
Some tips for solving integration questions effectively are to:
1. Familiarize yourself with the various integration techniques and their corresponding formulas.
2. Always check for any algebraic manipulations or substitutions that can simplify the integral.
3. Practice, practice, practice! The more you solve integration questions, the better you will become at identifying the right technique and applying it correctly.
4. Don't panic if you get stuck. Take a break and come back to the problem with a fresh perspective.
5. Use online resources or ask for help from peers or a teacher if you are struggling with a particular problem.