Evaluate the integral. int(-(24)/(x^(5))+10x^(4))dx Answer:

Given Integral Split: We are given the integral to evaluate: ∫(-(24/x^5) + 10x^4)dx We can split this integral into two separate integrals: ∫(-24/x^5)dx + ∫(10x^4)dxEvaluate First Term: Let's first evaluate the integral of the first term: ∫(-24/x^5)dx This is a simple power rule integration. We add 1 to the exponent and divide by the new exponent: -24 * ∫(x^(-5))dx = -24 * (x^(-5+1)/(-5+1)) + C = -24 * (x^(-4)/(-4)) + C = 6x^(-4) + CEvaluate Second Term: Now let's evaluate the integral of the second term: ∫(10x^4)dx Again, we use the power rule for integration: 10 * ∫(x^4)dx = 10 * (x^(4+1)/(4+1)) + C = 10 * (x^5/5) + C = 2x^5 + CCombine Results: Combining the results from both integrals, we get the final answer: 6x^(-4) + 2x^5 + C

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Calculus

Find indefinite integrals using the substitution

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Q. Evaluate the integral.

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\int\left(-\frac{24}{x^{5}}+10 x^{4}\right) d x

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Answer:

Given Integral Split: We are given the integral to evaluate: $\newline$ $\int(-\frac{24}{x^5} + 10x^4)\,dx$ $\newline$ We can split this integral into two separate integrals: $\newline$ $\int(-\frac{24}{x^5})\,dx + \int(10x^4)\,dx$
Evaluate First Term: Let's first evaluate the integral of the first term: $\newline$ $\int(-\frac{24}{x^5})dx$ $\newline$ This is a simple power rule integration. We add $1$ to the exponent and divide by the new exponent: $\newline$ $-24 \times \int(x^{-5})dx = -24 \times (\frac{x^{-5+1}}{-5+1}) + C$ $\newline$ $= -24 \times (\frac{x^{-4}}{-4}) + C$ $\newline$ $= 6x^{-4} + C$
Evaluate Second Term: Now let's evaluate the integral of the second term: $\newline$ $\int(10x^4)dx$ $\newline$ Again, we use the power rule for integration: $\newline$ $10 \times \int(x^4)dx = 10 \times (x^{4+1}/(4+1)) + C$ $\newline$ $= 10 \times (x^5/5) + C$ $\newline$ $= 2x^5 + C$
Combine Results: Combining the results from both integrals, we get the final answer: $6x^{-4} + 2x^5 + C$