Evaluate the indefinite integral. ∫−(6)xdx\int_{}^{}-\left(6\right)^{x}dx

− ( 6 ) x ln ⁡ 6 + C -\frac{\left(6\right)^{x}}{\ln6}+C

Evaluate the indefinite integral. ∫ − ( 6 ) x d x \int_{}^{}-\left(6\right)^{x}dx

this problem, we're asked to evaluate the indefinite integral of negative six to the power of x d x. Now we can see that this is a general exponential function, so we're gonna wanna apply this rule that we just learned here. Now looking at this, since my negative is outside of the six that is raised to the power of x, it itself is not being raised to the power of x. So I can actually just pull that negative out front. So I can rewrite this as negative, six negative the integral of six to the power of x d x. Then applying my rule here, this then becomes negative six to the power of x over the natural log of six and then, of course, plus my constant integration c and this is my final answer here. Let Let us know if you have any questions and let's keep practicing.

Evaluate the indefinite integral. ∫ − ( 6 ) x d x \int_{}^{}-\left(6\right)^{x}dx

− ( 6 ) x ln ⁡ 6 + C -\frac{\left(6\right)^{x}}{\ln6}+C

− x 6 x − 1 + C -x6^{x-1}+C

− 1 x + 1 6 x + 1 + C -\frac{1}{x+1}6^{x+1}+C

− ( 6 ) x + 1 ln ⁡ x + C -\frac{\left(6\right)^{x+1}}{\ln 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
  1h 8m
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.
$\int_{}^{} - {(6)}^{x} d x$

$- x 6^{x - 1} + C$

$- \frac{1}{x + 1} 6^{x + 1} + C$

$- \frac{{(6)}^{x}}{\ln 6} + C$

$- \frac{{(6)}^{x + 1}}{\ln x} + C$

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Verified step by step guidance

Step 1: Recognize that the integral involves an exponential function of the form $ 6^x $. The general formula for integrating $ a^x $ is $ \int a^x dx = \frac{a^x}{\ln a} + C $, where $ a > 0 $ and $ a \neq 1 $.

Step 2: Apply the formula to the given integral $ \int -6^x dx $. The negative sign can be factored out, so the integral becomes $ -\int 6^x dx $.

Step 3: Using the formula $ \int a^x dx = \frac{a^x}{\ln a} + C $, substitute $ a = 6 $. This gives $ \int 6^x dx = \frac{6^x}{\ln 6} + C $.

Step 4: Multiply the result by $ -1 $ to account for the negative sign. This gives $ -\frac{6^x}{\ln 6} + C $.

Step 5: Write the final expression for the indefinite integral as $ -\frac{6^x}{\ln 6} + C $, where $ C $ is the constant of integration.

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