Question

find the indefinite integral. (check by differentiation.)

$$\int ( 5 e ^ { z } + 11 ) d z$$

$$\int ( 5 e ^ { z } + 11 ) d z =$$

Explanation:

Step1: Apply integral sum rule

$$\int(5e^{z}+11)dz=\int5e^{z}dz+\int11dz$$

Step2: Use integral constant multiple rule

$$\int5e^{z}dz = 5\int e^{z}dz$$
Since $\int e^{z}dz=e^{z}+C_1$, then $5\int e^{z}dz = 5e^{z}+C_1$.
Also, $\int11dz=11\int dz$. As $\int dz=z + C_2$, then $11\int dz=11z + C_2$.

Step3: Combine results

$$\int(5e^{z}+11)dz=5e^{z}+11z + C$$ ($C = C_1 + C_2$)

Step4: Check by differentiation

Differentiate $5e^{z}+11z + C$ with respect to $z$.
Using the sum rule of differentiation: $\frac{d}{dz}(5e^{z}+11z + C)=\frac{d}{dz}(5e^{z})+\frac{d}{dz}(11z)+\frac{d}{dz}(C)$.
Since $\frac{d}{dz}(e^{z})=e^{z}$, $\frac{d}{dz}(z) = 1$ and $\frac{d}{dz}(C)=0$.
We get $5e^{z}+11\times1+0=5e^{z}+11$.

Answer:

$5e^{z}+11z + C$