Evaluate the integrals in Exercises 33–54.
∫(e^(3x) + 5e^(-x)) dx

Guida verificata passo dopo passo

Recognize that the integral is a sum of two separate integrals: \(\int (e^{3x} + 5e^{-x}) \, dx = \int e^{3x} \, dx + \int 5e^{-x} \, dx\).

Recall the integral formula for exponential functions: \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\), where \(a\) is a constant.

Apply the formula to the first integral: \(\int e^{3x} \, dx = \frac{1}{3} e^{3x} + C_1\).

Apply the formula to the second integral, factoring out the constant 5: \(\int 5e^{-x} \, dx = 5 \int e^{-x} \, dx = 5 \left(-e^{-x}\right) + C_2\).

Combine the results of both integrals and include a single constant of integration \(C\): \(\int (e^{3x} + 5e^{-x}) \, dx = \frac{1}{3} e^{3x} - 5 e^{-x} + C\).

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