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

Verified step by step guidance

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 \(C\): \(\int (e^{3x} + 5e^{-x}) \, dx = \frac{1}{3} e^{3x} - 5 e^{-x} + C\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration of Exponential Functions

Integrating exponential functions involves reversing differentiation rules. For functions like e^(ax), the integral is (1/a)e^(ax) + C, where a is a constant. This rule applies to each term separately in a sum.

Linearity of Integration

Integration is a linear operation, meaning the integral of a sum is the sum of the integrals. This allows us to split ∫(f(x) + g(x)) dx into ∫f(x) dx + ∫g(x) dx, simplifying the evaluation process.

Constant of Integration

When evaluating indefinite integrals, a constant of integration (C) must be added to represent all possible antiderivatives. This accounts for any constant term lost during differentiation.

Recommended video: