Questions: ∫(6 cos(3x) - 2e^x + 7/x + π^2) dx

Solution Steps

To solve the given integral, we need to integrate each term separately. The integral of a sum is the sum of the integrals. We will use the standard integration rules for each term.

We start by breaking down the integral into its individual components: \[ \int \left( 6 \cos(3x) - 2 e^x + \frac{7}{x} + \pi^2 \right) \, dx \]

We integrate each term separately using standard integration rules:

1. \(\int 6 \cos(3x) \, dx\)
2. \(\int -2 e^x \, dx\)
3. \(\int \frac{7}{x} \, dx\)
4. \(\int \pi^2 \, dx\)

1. For \(\int 6 \cos(3x) \, dx\): \[ \int 6 \cos(3x) \, dx = 2 \sin(3x) \]
2. For \(\int -2 e^x \, dx\): \[ \int -2 e^x \, dx = -2 e^x \]
3. For \(\int \frac{7}{x} \, dx\): \[ \int \frac{7}{x} \, dx = 7 \ln|x| \]
4. For \(\int \pi^2 \, dx\): \[ \int \pi^2 \, dx = \pi^2 x \]

We combine the results of the individual integrals: \[ \int \left( 6 \cos(3x) - 2 e^x + \frac{7}{x} + \pi^2 \right) \, dx = 2 \sin(3x) - 2 e^x + 7 \ln|x| + \pi^2 x + C \]

Final Answer