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

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

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

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

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

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

12 x 5 − 5 x + 1 ln ⁡ 5 + C 12x^5-\frac{5^{x+1}}{\ln5}+C

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
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5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
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10. Physics Applications of Integrals 3h 16m
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14. Sequences & Series5h 36m
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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.
$\int_{}^{} 3 x^{4} - {(5)}^{x} d x$

$\frac{3}{5} x^{5} - \frac{5^{x + 1}}{\ln 5} + C$

$\frac{3}{5} x^{5} - \frac{5^{x}}{\ln 5} + C$

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

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

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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, so rewrite it 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. Recall that the integral of an exponential function a^x 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 of the two integrals. The first term contributes (3/5)x^5, and the second term contributes -(5^x / ln(5)).

Step 5: Add the constant of integration, C, to the combined result. The final expression is (3/5)x^5 - (5^x / ln(5)) + C.