Find the most general antiderivative.
-$\int \left( 7 e ^ { 2 x } - 6 e ^ { - x } \right) d x$
A) $\frac { 7 } { 2 } e ^ { 2 x } + \frac { 1 } { 6 } e ^ { - x } + C$
B) $\frac { 7 } { 2 } e ^ { 2 x } - 6 e ^ { - x } + C$
C) $\frac { 2 } { 7 } e ^ { 2 x } + 6 e ^ { - x } + C$
D) $\frac { 7 } { 2 } e ^ { 2 x } + 6 e ^ { - x } + C$

Question

Quizplus · Accepted Answer

To find the antiderivative, we use the power rule of integration. For the first term, we have:

\begin{align*}
\int 7e^{2x}dx &= 7\int e^{2x}dx \
&= \frac{7}{2} e^{2x} + C_1
\end{align*}

For the second term, we use substitution. Let $u=-x$, then $du=-dx$. We have:

\begin{align*}
\int 6e^{-x}dx &= -6\int e^{u}du \
&= -6e^u + C_2 \
&= -6e^{-x} + C_2
\end{align*}

Combining both antiderivatives, we get:

$$\int \left( 7 e ^ { 2 x } - 6 e ^ { - x } ight) d x = \frac{7}{2} e^{2x} - 6e^{-x} + C$$

where $C=C_1+C_2$ is the constant of integration. Therefore, the best choice is D.

Find the Most General Antiderivative ∫(7e2x−6e−x)dx\int \left( 7 e ^ { 2 x } - 6 e ^ { - x } \right) d x∫(7e2x−6e−x)dx

Find the Most General Antiderivative $\int \left( 7 e ^ { 2 x } - 6 e ^ { - x } \right) d x$