Question

$$int (12e^x - 7^x + 7x^5) dx$$
a.) $12e^x + \frac{7^x}{ln 7} + \frac{7}{5}x^6 + c$
b.) $12e^x - \frac{7^x}{ln 7} + \frac{7}{6}x^6 + c$
c.) $12e^x + 7^x ln 7 + \frac{7}{6}x^6 + c$
d.) $12e^x - 7^x ln x + \frac{7}{5}x^6 + c$

Explanation:

Step1: Split the integral

$$\int(12e^x - 7^x + 7x^5)dx = \int12e^x dx - \int7^x dx + \int7x^5 dx$$

Step2: Integrate exponential term $12e^x$

$$\int12e^x dx = 12e^x$$

Step3: Integrate exponential term $7^x$

$$\int7^x dx = \frac{7^x}{\ln7}$$

Step4: Integrate polynomial term $7x^5$

$$\int7x^5 dx = 7\times\frac{x^{5+1}}{5+1} = \frac{7}{6}x^6$$

Step5: Combine results + constant

$$12e^x - \frac{7^x}{\ln7} + \frac{7}{6}x^6 + C$$

Answer:

b.) $12e^{x}-\frac{7^{x}}{\ln7}+\frac{7}{6}x^{6}+c$