Question

In Exercises \(1-6,\) find the indefinite integral. $$\int\left(2 e^{x}+\sec x \tan x-\sqrt{x}\right) d x$$

Short answer

The indefinite integral of \((2e^x + \sec{x}\tan{x} - \sqrt{x})\) is \[2e^x + \sec{x} - \frac{2}{3}x^{\frac{3}{2}} + C\]

Step-by-step solution

Identify Functions to be Integrated

The integrand is a sum of three functions. Thus, separate the integral into the sum of three integrals: \[ \int(2e^x + \sec{x}\tan{x} - \sqrt{x}) dx = 2\int e^x dx + \int \sec{x}\tan{x} dx - \int x^{\frac{1}{2}} dx \]

Apply Rules for Indefinite Integral

the integral of e^x is e^x, the integral of sec(x)tan(x) is sec(x), and we apply the power rule to the last integral. Therefore, we obtain:\[ 2\int e^x dx = 2e^x + C1 \]\[ \int \sec{x}\tan{x} dx = \sec{x} + C2 \]\[ \int x^{\frac{1}{2}} dx = \frac{2}{3} x^{\frac{3}{2}}+ C3 \]

Combine the Intermediary Solutions

We add the results of the integrals above, and remember, we can combine all the constants C1, C2, and C3 into one arbitrary constant, C, because it is also arbitrary and the sum or difference of arbitrary constants is arbitrary. Therefore, the solution to the original integral is:\[ \int(2e^x + \sec{x}\tan{x} - \sqrt{x}) dx = 2e^x + \sec{x} - \frac{2}{3}x^{\frac{3}{2}} + C \]