Evaluate the indefinite integral.∫2(13)xdx\int_{}^{}2\left(\frac13\right)^{x}dx

− 2 3 x ln ⁡ 3 + C -\frac{2}{3^{x}\ln3}+C

Evaluate the indefinite integral. ∫ 2 ( 1 3 ) x d x \int_{}^{}2\left(\frac13\right)^{x}dx

we're asked to evaluate the indefinite integral of two times one third to the power of x d x. Now looking at this here, I do have a general exponential function with a base of one third, so I want to go ahead and apply the integral rule here. Now two is just a constant, so I'm gonna keep that as is, and this becomes two times one third to the power of x over the natural log of one third, then, of course, plus our constant of integration c. Now because here I do have a one third being raised to the power of x and I'm taking the natural log of one third, I can actually simplify this or rewrite it using exponential and log properties. Now this as is is the correct answer, but depending on your textbook or your professor, they may prefer you simplify it down further. So let's rewrite this using our exponential and log properties. Now one third to the power of x, I can go ahead and raise each my raise both my numerator and denominator to that power of x. So in my numerator I have two times one to the power of x, and then I'm going to go ahead and put my three on the bottom there since it's on the bottom of a fraction. So that's three to the power of x and then times the natural log of three to the power of negative one. That's what I can rewrite one third as. So this is three to the power of negative one. Then one additional thing we can do here, this negative one using log properties that can pull that out front. So that makes this whole thing negative. And that's just the natural log of three now. Now one to the power of x is always going to be one, so I don't have to worry about that x. So on top here, I just have a two, and then all of this is negative since we pulled that negative one out front. And this is three to the power of x times the natural log of three. Now we also wanna keep our constant of integration c there and this is our final simplified answer having used exponential and log properties. Let us know if you have any questions and let's keep practicing.

Evaluate the indefinite integral. ∫ 2 ( 1 3 ) x d x \int_{}^{}2\left(\frac13\right)^{x}dx

− 2 3 x ln ⁡ 3 + C -\frac{2}{3^{x}\ln3}+C

− 2 x x + 1 4 + C -\frac{2x^{x+1}}{4}+C

− 2 x ln ⁡ 3 + C -\frac{2x}{\ln3}+C

− 2 x x + 1 3 x ln ⁡ 4 + C -\frac{2x^{x+1}}{3^{x}\ln4}+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
  1h 10m
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.
$\int_{}^{} 2 {(\frac{1}{3})}^{x} d x$

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

$- \frac{2 x^{x + 1}}{4} + C$

$- \frac{2 x}{\ln 3} + C$

$- \frac{2 x^{x + 1}}{3^{x} \ln 4} + C$

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Verified step by step guidance

Step 1: Recognize the integral to be evaluated: ∫2(1/3)^x dx. This is an exponential function with a base of 1/3. Recall that the general formula for integrating an exponential function of the form ∫a^x dx is (a^x / ln(a)) + C, where a > 0 and a ≠ 1.

Step 2: Rewrite the given integral to make it easier to work with. The integral can be expressed as ∫2 * (1/3)^x dx. Here, the constant 2 can be factored out of the integral, giving 2 * ∫(1/3)^x dx.

Step 3: Apply the formula for integrating exponential functions. For the term (1/3)^x, the base is 1/3. Using the formula, the integral of (1/3)^x is (1/3)^x / ln(1/3).

Step 4: Simplify the result. Note that ln(1/3) = ln(1) - ln(3) = -ln(3). Substituting this into the expression, the integral becomes -((1/3)^x / ln(3)). Don't forget to multiply by the constant 2 factored out earlier.

Step 5: Add the constant of integration, C, to account for the indefinite integral. The final expression is -2 * ((1/3)^x / ln(3)) + C.