Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.∫ (√x(2x⁶ - 4³√)dx

Accepted Answer

Rewrite the integral to simplify the expression. Break down the terms inside the integral: $$ \int \sqrt{x}(2x^6 - 4\sqrt[3]{x}) \, dx . $$Recall that $$ \sqrt{x} = x^{1/2} $$ and $$ \sqrt[3]{x} = x^{1/3} . $$Distribute $$ x^{1/2} $$ across the terms inside the parentheses. After distributing, the integral becomes $$ \int (2x^{6 + 1/2} - 4x^{1/2 + 1/3}) \, dx . $$Simplify the exponents by adding the powers: $$ 6 + 1/2 = 13/2 $$ and $$ 1/2 + 1/3 = 3/6 + 2/6 = 5/6 . $$The integral is now $$ \int (2x^{13/2} - 4x^{5/6}) \, dx . $$Apply the power rule for integration: $$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $$, where $$ n 
eq -1 . $$Integrate each term separately. For the first term, $$ 2x^{13/2} $$, the new exponent is $$ 13/2 + 1 = 15/2 $$, and the coefficient becomes $$ \frac{2}{15/2} = \frac{4}{15} . $$For the second term, $$ -4x^{5/6} $$, the new exponent is $$ 5/6 + 1 = 11/6 $$, and the coefficient becomes $$ \frac{-4}{11/6} = \frac{-24}{11} . $$Combine the results of the integration. The indefinite integral becomes $$ \frac{4}{15}x^{15/2} - \frac{24}{11}x^{11/6} + C $$, where $$ C  is $$the constant of integration. Check your work by differentiating the result. Differentiate $$ \frac{4}{15}x^{15/2} - \frac{24}{11}x^{11/6} + C $$ term by term using the power rule for differentiation: $$ \frac{d}{dx}x^n = nx^{n-1} . $$Verify that the derivative matches the original integrand $$ \sqrt{x}(2x^6 - 4\sqrt[3]{x}) .$$

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (√x(2x⁶ - 4³√)dx

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Key Concepts

Indefinite Integrals

Integration Techniques

Verification by Differentiation