Finding the second antiderivative given f(x) values

Question

Find $f$.

$f''(t)=7e^t + 3\sin(t)$, $f(0) = 0$ and $f(\pi)=0$

I found that:

$f'(t)= 7e^t + 3\cos(t) + C$

$f(t) = 7e^t - 3\sin(t) + Ct + D$

How do I know what equation to plug the $x$-values into?

Answer 1

$$ f''(t)=7e^t + 3\sin{t}\Longleftrightarrow \int\left(7e^t + 3\sin{t}\right)\,dt=f'(t)\implies\\ f'(t)=\int\left(7e^t + 3\sin{t}\right)\,dt=7e^t - 3\cos{t}+C_1 $$

Use the same logic to find the original function itself (in fact, it's going to be a family of functions because of the constants that appear as a result of the integration process):

$$ f'(t)=7e^t - 3\cos{t}+C_1\Longleftrightarrow \int\left(7e^t - 3\cos{t}+C_1\right)\,dt=f(t)\implies\\ f(t)=\int\left(7e^t - 3\cos{t}+C_1\right)\,dt=7e^t - 3\sin{t}+C_1t+C_2 $$

Now, use the information you're given as part of the problem statement to find out what these constants are equal to:

$$ f(0)=0\implies 7e^0 - 3\sin{0}+C_1\cdot 0+C_2=0\implies C_2=-7\\ f(\pi)=0\implies 7e^\pi - 3\sin{\pi}+C_1\pi-7=0\implies C_1=\frac{7(1-e^\pi)}{\pi} $$

So, the original function $f(t)$ is then:

$$ f(t)=7e^t - 3\sin{t}+\frac{7(1-e^\pi)}{\pi}-7. $$

Answer 2

If you have $$ f(t) = 7e^t-3\sin(t)+Ct+D$$ (although I would check the signs on the trig functions) and you know $f(0)=0$ and $f(\pi)=0$. The above is what you plug into!
$$ f(0) = 7e^0-3\sin(0)+C(0)+D = 0 $$ and $$ f(\pi) = 7e^\pi-3\sin(\pi)+C(\pi)+D = 0 $$ This is a system of two equations with two unknowns ($C$ and $D$), which you can solve.

Answer 3

You made a mistake in taken derivative:

$$ f^{''}(t)=7e^{t}+3\sin(t) $$ $$ f^{'}(t)=7e^{t}-3\cos(t)+C$$ $$ f(t)=7e^{t}-3\sin(t)+Ct+D$$

$$ f(0)=0 : 7-0+0+D=0 : D=-7$$

$$ f(\pi)=0 : 7e^{\pi}-0+C\pi+-7=0 : C=\frac{7-7e^{\pi}}{\pi}$$