find the indefinite integral. (check by differentiation.) ∫(8e^z + 6) dz ∫(8e^z + 6) dz =

Answer:

$8e^{z}+6z + C$

Explanation:

Step1: Apply integral sum - rule

$\int(8e^{z}+6)dz=\int8e^{z}dz+\int6dz$

Step2: Integrate $\int8e^{z}dz$

Since $\int e^{z}dz = e^{z}$, then $\int8e^{z}dz = 8e^{z}$

Step3: Integrate $\int6dz$

$\int6dz=6z$

Step4: Add the results and the constant of integration

$8e^{z}+6z + C$

Step5: Check by differentiation

Differentiate $8e^{z}+6z + C$. Using the sum - rule of differentiation, $\frac{d}{dz}(8e^{z}+6z + C)=\frac{d}{dz}(8e^{z})+\frac{d}{dz}(6z)+\frac{d}{dz}(C)$. Since $\frac{d}{dz}(e^{z}) = e^{z}$, $\frac{d}{dz}(8e^{z}) = 8e^{z}$, $\frac{d}{dz}(6z)=6$ and $\frac{d}{dz}(C) = 0$, we get $8e^{z}+6$, which is the original integrand.