Question

secret lab iii ... do not enter :/

Answer 1

\begin{align}
6)\ F(x)&= \int\ 8x^9-3x^6+12x^3\ dx\\\\
&=\frac{4}{5}x^{10} -\frac{3}{7}x^7 +3x^4+C\\\\
10)\ F(x) &=\int\ x^{3/4}+x^{4/3}\ dx\\\\
&=\frac{4x^{7/4}+3x^{7/3}}{7}+C\\\\
12)\ F(x)&=\int\ 5x^{-6}-4x^{-3}+2\ dx\\\\
&=x^{-5}+2x^{-2}+2x+C\\\\
&=-\frac{1}{x^5}+\frac{2}{x^2}+2x+C\\\\
18)\ F(x)&=\int\ 2x^{1/2}+6\ cos(x)\ dx\\\\
&=\frac{4}{3}x^{3/2}+6\ sin(x)+C\\\\
20)\ F(x)&=\int\ \frac{2+x^2}{1+x^2}dx\\\\
&=\int\ \frac{1}{1+x^2}+1\ dx\\\\
&=tan^{-1}(x)+x+C\\\\
34)\ F(x)&=\int\ \frac{x^2-1}{x}dx;\ F(1)=\frac{1}{2},\ F(-1)=0\\\\
&=\int\ x-\frac{1}{x}\ dx\\\\
&=\frac{x^2}{2}+ln|x|+C
\end{align}

Answer 2

\begin{align}
34)\ F(x)&=\int\ \frac{x^2-1}{x}dx;\ F(1)=\frac{1}{2},\ F(-1)=0\\\\
&=\int\ x-\frac{1}{x}\ dx\\\\
&=\frac{x^2}{2}+ln|x|+C\\\\\\\\
F(x)&=\left\{\begin{array}l\frac{x^2}{2} -\ln|x|\ \ \ \ \ \ \ \ ; & \text{ if } x>0 \\ \\\frac{x^2}{2} - \ln|x| -\frac{1}{2}; & \text{ if } x<0 \end{array}\right.
\end{align}

Answer 3

\begin{align}
38)\ f''(x)&=4-6x-40x^3;\ f'(0)=1,\ f(0)=2\\\\
f'(x)&=4x-3x^2-10x^4+1\\\\
f(x)&=2x^2-x^3-2x^5+x+2\\\\
44)\ f''(x)&=2e^x+3sin(x);\ f(0)=0,\ f(\pi)=0\\\\
f'(x)&=2e^x-3cos(x)+C_1\\\\
f(x)&=2e^x-3sin(x)+C_1x+C_2\\\\
f(0)&=2e^0-3sin(0)+C_1(0)+C_2=0\\\\
0&=2+C_2\\\\
-2&=C_2\\\\
f(\pi)&=2e^x-3sin(x)+C_1x-2=0\\\\
0&=2e^\pi-3sin(\pi)+C_1\pi-2\\\\
C_1&=2\frac{(1-e^\pi)}{\pi}\\\\
f(x)&=2e^x-3sin(x)+2x\frac{(1-e^\pi)}{\pi}-2\\\\
\end{align}

Answer 4

\begin{align}
20) F(x)&=\\\\
&=\int\ \frac{1+1+x^2}{1+x^2}dx\\\\
&=\int\ \frac{1}{1+x^2}+\frac{1+x^2}{1+x^2}dx\\\\
\end{align}

Answer 5

im pretty sure i fixed all the typos in the final draft that i printed out after this one :)

Answer 6

wow great