Q. Calculate the integral and write the answer in simplest form.∫(−2x−3−4x4)dxAnswer:
Identify Integral: Identify the integral to be solved.We need to find the indefinite integral of the function −2x−3−4x4 with respect to x.
Break Down Parts: Break down the integral into simpler parts.The integral of a sum of functions is the sum of the integrals of each function. Therefore, we can write:∫(−2x−3−4x4)dx=∫(−2x−3)dx+∫(−4x4)dx
Integrate First Part: Integrate the first part of the function.To integrate −2x−3, we use the power rule of integration, which states that ∫xndx=n+1xn+1+C, where n=−1.\int(\(-2x^{−3})\,dx = −2 \times \int x^{−3}\,dx = −2 \times \left(\frac{x^{−3+1}}{−3+1}\right) + C = −2 \times \left(\frac{x^{−2}}{−2}\right) + C = x^{−2} + C
Integrate Second Part: Integrate the second part of the function.Now we integrate −4x4 using the same power rule.\int(\(-4x^{4})dx = −4 \times \int x^{4}dx = −4 \times \left(\frac{x^{4+1}}{4+1}\right) + C = −4 \times \left(\frac{x^{5}}{5}\right) + C = -\left(\frac{4}{5}\right)x^{5} + C
Combine Results: Combine the results from Step 3 and Step 4.The combined result of the integrals is:x−2−54x5+C
Simplify Final Form: Simplify the result and express it in simplest form.The simplest form of the result is:x21−(54)x5+C
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