Evaluate the indefinite integral.
${\int }_{}^{}3x^4-{\left(5\right)}^{x}dx$

• A. $\frac{3}{5}{x}^5-\frac{5^{x+1}}{\ln 5}+C$
• B. $\frac{3}{5}{x}^5-\frac{5^x}{\ln 5}+C$
• C. $12x^5-\frac{5^x}{\ln 5}+C$
• D. $12x^5-\frac{5^{x+1}}{\ln 5}+C$

Answer: (B)

Verified step by step guidance

1. 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.
2. 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.
3. 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)).
4. 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)).
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.