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

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

Step3: Use basic integral formulas

Since $\int e^{z}dz = e^{z}+C_1$ and $\int dz=z + C_2$
$$12\int e^{z}dz+\int5dz=12e^{z}+5z + C$$ (where $C = C_1 + C_2$)

Step4: Check by differentiation

Differentiate $12e^{z}+5z + C$ with respect to $z$
Using the sum rule of differentiation $\frac{d}{dz}(u + v+w)=\frac{du}{dz}+\frac{dv}{dz}+\frac{dw}{dz}$
$\frac{d}{dz}(12e^{z})=12e^{z}$, $\frac{d}{dz}(5z)=5$, $\frac{d}{dz}(C) = 0$
So $\frac{d}{dz}(12e^{z}+5z + C)=12e^{z}+5$

Answer:

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