Evaluate the indefinite integral.∫3x4−(5)xdx\int_{}^{}3x^4-\left(5\right)^{x}dx∫3x4−(5)xdx

3 5 x 5 − 5 x ln ⁡ 5 + C \frac35x^5-\frac{5^{x}}{\ln5}+C 5 3 x 5 − l n 5 5 x + C

Evaluate the indefinite integral. ∫ 3 x 4 − ( 5 ) x d x \int_{}^{}3x^4-\left(5\right)^{x}dx ∫ 3 x 4 − ( 5 ) x d x

we're asked to evaluate the indefinite integral of three x to the power of four minus five to the power of x d x. Now here, because in my integrand I have two functions being subtracted, I really just want to take the antiderivative of each of them individually and subtract them from each other. Now I could go ahead and break this out into two separate integrals, but I'm gonna go ahead and do this all at once here. If you have any questions, feel free to let us know in the comments. Now focusing on that first term here, the integral of three x to the power of four. Now three is just a constant, so I'm taking that three and I'm multiplying it by the antiderivative of x to the power of four, which is one fifth x to the power of five. So this is really three fifths x to the power of five, and then subtracting off now my integral of five to the power of x, we wanna use the rule that we just learned here for our general exponential functions. So this is then five to the power of x over the natural log of five. And then we wanna go ahead and add on our constant integration c here because this is an indefinite integral. So this is my final answer. Again, if you have questions, feel free to let us know in the comments. I'll see you in the next one.

Evaluate the indefinite integral. ∫ 3 x 4 − ( 5 ) x d x \int_{}^{}3x^4-\left(5\right)^{x}dx ∫ 3 x 4 − ( 5 ) x d x

3 5 x 5 − 5 x ln ⁡ 5 + C \frac35x^5-\frac{5^{x}}{\ln5}+C 5 3 x 5 − l n 5 5 x + C

3 5 x 5 − 5 x + 1 ln ⁡ 5 + C \frac35x^5-\frac{5^{x+1}}{\ln5}+C 5 3 x 5 − l n 5 5 x + 1 + C

12 x 5 − 5 x ln ⁡ 5 + C 12x^5-\frac{5^{x}}{\ln5}+C 12 x 5 − l n 5 5 x + C

12 x 5 − 5 x + 1 ln ⁡ 5 + C 12x^5-\frac{5^{x+1}}{\ln5}+C 12 x 5 − l n 5 5 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.
$\int_{}^{}3x^4-\left(5\right)^{x}dx$

$\frac35x^5-\frac{5^{x+1}}{\ln5}+C$

$\frac35x^5-\frac{5^{x}}{\ln5}+C$

$12x^5-\frac{5^{x}}{\ln5}+C$

$12x^5-\frac{5^{x+1}}{\ln5}+C$

Verified step by step guidance

Step 1: Break the integral into two separate terms for easier evaluation. The given integral is ∫(3x^4 - 5^x) dx. This can be rewritten as ∫3x^4 dx - ∫5^x dx.

Step 2: Evaluate the first term, ∫3x^4 dx. Use the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1. Here, n = 4, so the integral becomes (3/5)x^5.

Step 3: Evaluate the second term, ∫5^x dx. For exponential functions of the form a^x, the integral is (a^x / ln(a)) + C, where a > 0 and a ≠ 1. Here, a = 5, so the integral becomes (5^x / ln(5)).

Step 4: Combine the results from Step 2 and Step 3. The integral becomes (3/5)x^5 - (5^x / ln(5)) + C.

Step 5: Verify the final expression matches the correct answer format. The final result is (3/5)x^5 - (5^x / ln(5)) + C, which is consistent with the correct answer provided in the problem.

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