Calculate the integral and write the answer in simplest form. int(6x^(3)+5-x^(-3))dx Answer:

Given Integral: We are given the integral of a polynomial function with a negative exponent term: int(6x^(3) + 5 - x^(-3))dx To solve this, we will integrate each term separately.Integrating 6x^3: First, we integrate the term 6x^3: int(6x^(3))dx = 6 * int(x^(3))dx = 6 * (x^(3+1)/(3+1)) = 6 * (x^4/4) = (3/2)x^4Integrating Constant Term: Next, we integrate the constant term 5: int(5)dx = 5 * int(1)dx = 5xIntegrating -x^(-3): Finally, we integrate the term -x^(-3): int(-x^(-3))dx = -int(x^(-3))dx = - (x^(-3+1)/(-3+1)) = - (x^(-2)/(-2)) = (1/2)x^(-2) = (1/2)x^(-2)Combining Integrated Terms: Now, we combine all the integrated terms and add the constant of integration C: (3/2)x^4 + 5x + (1/2)x^(-2) + C

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Find indefinite integrals using the substitution

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Q. Calculate the integral and write the answer in simplest form.

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\int\left(6 x^{3}+5-x^{-3}\right) d x

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Answer:

Given Integral: We are given the integral of a polynomial function with a negative exponent term: $\int(6x^{3} + 5 - x^{-3})dx$ To solve this, we will integrate each term separately.
Integrating $6x^3$ : First, we integrate the term $6x^3$ : $\int(6x^{3})dx = 6 \times \int(x^{3})dx = 6 \times \left(\frac{x^{3+1}}{3+1}\right) = 6 \times \left(\frac{x^4}{4}\right) = \left(\frac{3}{2}\right)x^4$
Integrating Constant Term: Next, we integrate the constant term $5$ : $\int(5)\,dx = 5 \times \int(1)\,dx = 5x$
Integrating $-x^{-3}$ : Finally, we integrate the term $-x^{-3}$ : $\int(-x^{-3})dx = -\int(x^{-3})dx = - \left(\frac{x^{-3+1}}{-3+1}\right) = - \left(\frac{x^{-2}}{-2}\right) = \left(\frac{1}{2}\right)x^{-2} = \left(\frac{1}{2}\right)x^{-2}$
Combining Integrated Terms: Now, we combine all the integrated terms and add the constant of integration $C$ : $\newline$ $\left(\frac{3}{2}\right)x^4 + 5x + \left(\frac{1}{2}\right)x^{-2} + C$