We are integrating f(x) = 9cos(9x) + 3x²: [tex]\int\ {9cos(9x)+3x^{2} } \, dx[/tex]
a) Apply the sum rule
[tex]\int\9cos(9x)} \, dx +\int\ 3x^{2} } \, dx[/tex]
b) Calculate each antiderivative
First integral
[tex]\int\ {9cos(9x)} \, dx[/tex]
1. Take out the constant
[tex]9\int\ {cos(9x)} \, dx[/tex]
2. Apply u-substitution, where u is 9x
[tex]9\int\ {cos(u)\frac{1}{9} } du[/tex]
3. Take out the constant (again)
[tex]9*\frac{1}{9} \int{cos(u)} du[/tex]
4. Take the common integral of cos, which is sin
[tex]9*\frac{1}{9}sin(u)}[/tex]
5. Substitute the original function back in for u and simplify[tex]9*\frac{1}{9} sin(9x) = sin(9x)[/tex]
6. Always remember to add an arbitrary constant, C, at the end
[tex]sin(9x) + C[/tex]
Second integral
[tex]\int3x^{2} } \, dx[/tex]
1. Take out the constant
[tex]3\int{x^{2} } \, dx[/tex]
2. Apply the power rule, [tex]\int{x}^{a} \, dx =\frac{x^{a+1} }{a+1}[/tex], where a is your exponent
⇒ [tex]3*\frac{x^{2+1} }{2+1} = x^{3}[/tex]
3. Add the arbitrary constant
[tex]x^{3} + C[/tex]
c) Add the integrals
sin(9x) + C + x³ + C = sin(9x) + x³ + C
Notice the two arbitrary constants. Since we do not know what either constant is, we can combine them into one arbitrary constant.
F(x) = sin(9x) + x³ + C
Answer:
[tex]g(x)=sin(9x)+x^3+C[/tex], choose a constant that is any number you want for C
Step-by-step explanation:
