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Chapter 4: Problem 48

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(e^{2 t}+2 \sqrt{t}\right) d t$$

Short Answer

Expert verified
The indefinite integral of the given function with respect to t is: $$\int\left(e^{2t} + 2\sqrt{t}\right)dt = \frac{1}{2}e^{2t} + \frac{4}{3}t^{\frac{3}{2}} + C$$ where C is the constant of integration.

Step by step solution

01

Break down the integral into two separate integrals

We can split the given integral into two parts as follows: $$\int\left(e^{2t} + 2\sqrt{t}\right)dt = \int e^{2t}dt + \int 2\sqrt{t}dt$$ Now, we will integrate each part separately.
02

Integrate the first part - e^(2t)

To find the integral of the first part, we will use the rule \(\int e^{ax}dx = \frac{1}{a}e^{ax} + C\). For our question, \(a = 2\). So, we have: $$\int e^{2t}dt = \frac{1}{2}e^{2t} + C_1$$
03

Integrate the second part - 2sqrt(t)

The given function is \(2\sqrt{t}\) which can also be written as \(2t^{\frac{1}{2}}\). Now, we integrate this function using the power rule: \(\int x^n dx = \frac{x^{n+1}}{n+1}+C\). In our case, \(n = \frac{1}{2}\). $$\int 2t^{\frac{1}{2}}dt = 2\int t^{\frac{1}{2}}dt = 2\left(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right)+C_2 = \frac{4}{3}t^{\frac{3}{2}}+C_2$$
04

Combine the results

Now, we just need to combine our results from the first and second parts to get the overall indefinite integral: $$\int\left(e^{2t} + 2\sqrt{t}\right)dt = \frac{1}{2}e^{2t} + \frac{4}{3}t^{\frac{3}{2}} + C$$ where \(C = C_1 + C_2\) is the constant of integration.
05

Verify by differentiation

Differentiating our integral with respect to \(t\), we should get the original function back: $$\frac{d}{dt}\left(\frac{1}{2}e^{2t} + \frac{4}{3}t^{\frac{3}{2}} + C\right) = e^{2t} + 2\sqrt{t}$$ So, the indefinite integral is correct, and we have solved the problem.

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