Question

From Chapter 4.9, find the exact solution to

f^' (x)=-4 sin⁡(x)-6 cos⁡(x)+4 if f(0)=12 this means to find the value of C when you integrate given the condition above!

Answer 1

Answer:

$F(x)=4cos(x)-6sin(x)+4x+8$

Step-by-step explanation:

1. Write the expression.

$f'(x)=-4sin(x)-6cos(x)+4$

2. Write the expression for the integral.

$\int\((-4sin(x)-6cos(x)+4) \, dx$

3. Separate into multiple integrals.

$\int\((-4sin(x)) \, dx+\int\((-6cos(x)) \, dx+\int\((4) \, dx$

4. Solve each integral.

• Check the attached image for a better understanding of these results.

$\int\((-4sin(x)) \, dx=\\ \\-4\int\((sin(x)) \, dx=\\ \\-4(-cos(x))+C=\\ \\4cos(x)+C$

---------------------------------------------------------------------------------------------------------

$\int\((-6cos(x)) \, dx=\\ \\-6\int\((cos(x)) \, dx=\\\\-6(sin(x))+C=\\ \\-6sin(x)+C$

---------------------------------------------------------------------------------------------------------

$\int\((4) \, dx=\\ \\4x+C$

5. Sum up all the integrals.

$(4cos(x))+(-6sin(x))+(4x)+C\\ \\4cos(x)-6sin(x)+4x+C$

6. Write in standard form (solved for y).

$y=4cos(x)-6sin(x)+4x+C$

7. Substitute the given values and solve for C.

$12=4cos(0)-6sin(0)+4(0)+C\\ \\4cos(0)-6sin(0)+4(0)+C=12\\ \\C=12-4cos(0)+6sin(0)-4(0)\\ \\C=12-4(1)+6(0)-4(0)\\ \\C=12-4\\ \\C=8$

8. Express your result.

$F(x)=4cos(x)-6sin(x)+4x+8$

---------------------------------------------------------------------------------------------------------

How to verify the result?

If the result is correct, then F(0)=12. Let's test it!

$F(0)=4cos(0)-6sin(0)+4(0)+8=12$. Hence, the answer is correct.

---------------------------------------------------------------------------------------------------------

The graph.

I want to share the graph of this function, it looks pretty interesting. Check it out in attached image 2.

Answer 2

Answer:

$\text{f}(x)=4\cos(x)-6\sin(x)+4x+8$

Step-by-step explanation:

Given:

• $\text{f}\:'(x)=-4 \sin(x)-6 \cos (x)+4$

• $\text{f}(0)=12$

Fundamental Theorem of Calculus

$\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))$

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

To find the function f(x), integrate f'(x) and use f(0) = 12 to find the value of the constant.

$\begin{aligned}\displaystyle \int (-4 \sin(x)-6 \cos (x)+4)\:\:\text{d}x & = \int -4 \sin(x)\:\:\text{d}x -\int 6 \cos(x)\:\:\text{d}x+\int 4\:\:\text{d}x \\\\& =-4 \int \sin(x)\:\:\text{d}x -6\int \cos(x)\:\:\text{d}x +\int 4\:\:\text{d}x\\\\& = -4 \cdot -\cos(x)-6 \cdot \sin(x)+4x+\text{C}\\\\& = 4\cos(x)-6\sin(x)+4x+\text{C}\end{aligned}$

To find the value of C, substitute x = 0 into the function and set it to 12:

$\begin{aligned}\text{f}(0) & =12\\\implies 4 \cos(0)-6\sin(0)+4(0)+\text{C} & =12\\4-0+0+\text{C} & =12\\4+\text{C} & = 12\\\text{C} & =8\end{aligned}$

Finally, substitute the found value of C into the equation:

$\text{f}(x)=4\cos(x)-6\sin(x)+4x+8$

Rules of Integration

$\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\ \displaystyle \int ax^n\:\text{d}x=a \int x^n \:\text{d}x \end{minipage}}$

If the terms are multiplied by constants, take them outside the integral.

$\boxed{\begin{minipage}{5 cm}\underline{Integrating a constant}\\\\ \displaystyle \int n\:\text{d}x=nx+\text{C} \\(where n is any constant value)\end{minipage}}$

Just add an x to the constant.

$\boxed{\begin{minipage}{6 cm}\underline{Integrating Trigonometric functions}\\\\ \displaystyle \int \sin(x)\:\text{d}x=- \cos (x)+\text{C}\\\\ \int \cos (x)\:\text{d}x=\sin(x)+\text{C} \\\end{minipage}}$

Learn more about integration here:

https://brainly.com/question/27805589

https://brainly.com/question/28155016