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

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

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

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 d x

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

− x 6 x − 1 + C -x6^{x-1}+C − x 6 x − 1 + C

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

− ( 6 ) x + 1 ln ⁡ x + C -\frac{\left(6\right)^{x+1}}{\ln x}+C − l n x ( 6 ) x + 1 + C

10. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions

Business Calculus

10. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions

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Multiple Choice

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

$-x6^{x-1}+C$

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

$-\frac{\left(6\right)^{x}}{\ln6}+C$

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

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 for exponential integration to $ -6^x $. Since the base $ a $ is $ 6 $, substitute $ a = 6 $ into the formula. The integral becomes $ \int -6^x dx = -\frac{6^x}{\ln(6)} + C $.

Step 3: Simplify the expression. The negative sign remains outside the fraction, and $ \ln(6) $ is the natural logarithm of $ 6 $. The result is $ -\frac{6^x}{\ln(6)} + C $.

Step 4: Verify the solution by differentiating $ -\frac{6^x}{\ln(6)} + C $. Using the chain rule, the derivative of $ \frac{6^x}{\ln(6)} $ is $ 6^x $, and the negative sign ensures the derivative matches the original integrand $ -6^x $.

Step 5: Conclude that the indefinite integral of $ -6^x $ is $ -\frac{6^x}{\ln(6)} + C $, where $ C $ is the constant of integration.

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