Question

find the indefinite integral. (check by differentiation.)

intleft(12 e^{z}+5
ight) d z

intleft(12 e^{z}+5
ight) d z=

Explanation:

Step1: Apply integral sum rule

$$\int(12e^{z}+5)dz=\int12e^{z}dz+\int5dz$$

Step2: Apply constant multiple rule for integrals

For $\int12e^{z}dz$, since $\int kf(z)dz = k\int f(z)dz$ ($k = 12$, $f(z)=e^{z}$), we have $12\int e^{z}dz$. And for $\int5dz$, $k = 5$, $f(z)=1$.
$$\int12e^{z}dz+\int5dz=12\int e^{z}dz+5\int1dz$$

Step3: Use basic integral formulas

We know that $\int e^{z}dz=e^{z}+C_1$ and $\int1dz=z + C_2$.
So $12\int e^{z}dz+5\int1dz=12e^{z}+5z + C$ ($C = C_1 + C_2$ is the constant of integration)

Step4: Check by differentiation

Differentiate $y = 12e^{z}+5z + C$ using sum rule $(u + v+w)'=u'+v'+w'$ (here $u = 12e^{z}$, $v = 5z$, $w = C$).
The derivative of $u = 12e^{z}$ is $u'=12e^{z}$ (since $\frac{d}{dz}(e^{z})=e^{z}$ and using constant - multiple rule), the derivative of $v = 5z$ is $v'=5$ (since $\frac{d}{dz}(z)=1$ and using constant - multiple rule), and the derivative of $w = C$ is $w' = 0$ (since the derivative of a constant is $0$).
So $y'=12e^{z}+5$, which is the integrand.

Answer:

$12e^{z}+5z + C$