Evaluate the indefinite integral.∫(x−ex)dx\int_{}^{}\left(\sqrt{x}-e^{x}\right)dx

2 3 x 3 2 − e x + C \frac23x^{\frac32}-e^{x}+C

Evaluate the indefinite integral. ∫ ( x − e x ) d x \int_{}^{}\left(\sqrt{x}-e^{x}\right)dx

we're asked to evaluate the indefinite integral of the square root of x minus e to the x d x. Now looking at this integral, I know that the square root of x is the same thing as x to the power of one half, so I can go ahead and use the general power rule to find the antiderivative of this first term. So using my general power rule here, that makes this two thirds times x to the power of three halves. Then subtracting off this next anti the antiderivative of my next term here e to the x. I know that is also still e to the x, so subtracting e to the x and then of course adding my constant of integration c and this gives me my final answer. Two thirds x to the power of three halves minus e to the power of x plus c. Let us know if you have any questions and let's keep going.

Evaluate the indefinite integral. ∫ ( x − e x ) d x \int_{}^{}\left(\sqrt{x}-e^{x}\right)dx

2 3 x 3 2 − e x + C \frac23x^{\frac32}-e^{x}+C

2 3 x 2 3 − e x + C \frac23x^{\frac23}-e^{x}+C

2 3 x 3 2 − x e x + C \frac23x^{\frac32}-xe^{x}+C

2 3 x 2 3 − 1 x e x + C \frac23x^{\frac23}-\frac{1}{x}e^{x}+C

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
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10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
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16. Parametric Equations & Polar Coordinates7h 58m

11. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions

Calculus

11. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions

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Multiple Choice

Evaluate the indefinite integral.
$\int_{}^{} (\sqrt{x} - e^{x}) d x$

$\frac{2}{3} x^{\frac{2}{3}} - e^{x} + C$

$\frac{2}{3} x^{\frac{3}{2}} - x e^{x} + C$

$\frac{2}{3} x^{\frac{3}{2}} - e^{x} + C$

$\frac{2}{3} x^{\frac{2}{3}} - \frac{1}{x} e^{x} + C$

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Verified step by step guidance

Step 1: Break the integral into two separate terms for easier handling. The given integral is ∫(√x - e^x) dx, which can be rewritten as ∫√x dx - ∫e^x dx.

Step 2: Focus on the first term, ∫√x dx. Rewrite √x as x^(1/2) to make it easier to apply the power rule for integration. The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1.

Step 3: Apply the power rule to ∫x^(1/2) dx. Add 1 to the exponent (1/2 + 1 = 3/2) and divide by the new exponent (3/2). This gives (2/3)x^(3/2).

Step 4: Now, handle the second term, ∫e^x dx. The integral of e^x is simply e^x, as the derivative of e^x is itself.

Step 5: Combine the results from both terms. The final expression for the indefinite integral is (2/3)x^(3/2) - e^x + C, where C is the constant of integration.