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Calculate the integral and write the answer in simplest form.

int(-2x^(-3)-4x^(4))dx
Answer:

Calculate the integral and write the answer in simplest form.\newline(2x34x4)dx \int\left(-2 x^{-3}-4 x^{4}\right) d x \newlineAnswer:

Full solution

Q. Calculate the integral and write the answer in simplest form.\newline(2x34x4)dx \int\left(-2 x^{-3}-4 x^{4}\right) d x \newlineAnswer:
  1. Identify Integral: Identify the integral to be solved.\newlineWe need to find the indefinite integral of the function 2x34x4-2x^{-3}-4x^{4} with respect to xx.
  2. Break Down Parts: Break down the integral into simpler parts.\newlineThe integral of a sum of functions is the sum of the integrals of each function. Therefore, we can write:\newline(2x34x4)dx=(2x3)dx+(4x4)dx\int(-2x^{-3}-4x^{4})dx = \int(-2x^{-3})dx + \int(-4x^{4})dx
  3. Integrate First Part: Integrate the first part of the function.\newlineTo integrate 2x3-2x^{-3}, we use the power rule of integration, which states that xndx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, where n1n \neq -1.\newline\int(\(-2x^{3-3})\,dx = 2-2 \times \int x^{3-3}\,dx = 2-2 \times \left(\frac{x^{3-3+11}}{3-3+11}\right) + C = 2-2 \times \left(\frac{x^{2-2}}{2-2}\right) + C = x^{2-2} + C
  4. Integrate Second Part: Integrate the second part of the function.\newlineNow we integrate 4x4-4x^{4} using the same power rule.\newline\int(\(-4x^{44})dx = 4-4 \times \int x^{44}dx = 4-4 \times \left(\frac{x^{44+11}}{44+11}\right) + C = 4-4 \times \left(\frac{x^{55}}{55}\right) + C = -\left(\frac{44}{55}\right)x^{55} + C
  5. Combine Results: Combine the results from Step 33 and Step 44.\newlineThe combined result of the integrals is:\newlinex245x5+Cx^{-2} - \frac{4}{5}x^{5} + C
  6. Simplify Final Form: Simplify the result and express it in simplest form.\newlineThe simplest form of the result is:\newline1x2(45)x5+C\frac{1}{x^2} - \left(\frac{4}{5}\right)x^{5} + C