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Chapter 4: Problem 109

Determine the following indefinite integrals. Check your work by differentiation. $$\int \sqrt{x}\left(2 x^{6}-4 \sqrt[3]{x}\right) d x$$

Short Answer

Expert verified
The indefinite integral of the given function is: $$\int \sqrt{x}\left(2 x^{6}-4 \sqrt[3]{x}\right) dx = \frac{4}{15}x^{\frac{15}{2}} - \frac{24}{11}x^{\frac{11}{6}} + C$$

Step by step solution

01

Rewrite the given function in a more simplified form

Let's rewrite \(\sqrt{x}\) and \(\sqrt[3]{x}\) as \(x^{\frac{1}{2}}\) and \(x^{\frac{1}{3}}\), respectively. The given function can then be rewritten as: $$\int x^{\frac{1}{2}}\left(2 x^{6}-4 x^{\frac{1}{3}}\right)dx$$
02

Distribute the first term inside the parentheses

Now, we distribute \(x^\frac{1}{2}\) inside the parentheses: $$\int (2x^{\frac{1}{2}}x^6 - 4x^{\frac{1}{2}}x^{\frac{1}{3}})dx$$
03

Simplify the exponents

We can combine the exponents with the same base (x) by adding the exponents: $$\int (2x^{\frac{13}{2}} - 4x^{\frac{5}{6}})dx$$
04

Integrate each term separately

Now, we will integrate each term with respect to x: $$\int 2x^{\frac{13}{2}}dx - \int 4x^{\frac{5}{6}}dx$$ To do this, we will use the power rule for integration, which is: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ Applying this rule, we get: $$\frac{2x^{\frac{15}{2}}}{\frac{15}{2}} - \frac{4x^{\frac{11}{6}}}{\frac{11}{6}} +C$$
05

Simplify the final result

Simplify the terms and rewrite the exponents in radical notation: $$\frac{4}{15}x^{\frac{15}{2}} - \frac{24}{11}x^{\frac{11}{6}} + C$$ Thus, the indefinite integral of the given function is: $$\int \sqrt{x}\left(2 x^{6}-4 \sqrt[3]{x}\right) dx = \frac{4}{15}x^{\frac{15}{2}} - \frac{24}{11}x^{\frac{11}{6}} + C$$
06

Check our work by differentiation

Differentiate our result with respect to x, using the power rule for differentiation: $$\frac{d}{dx}\left( \frac{4}{15}x^{\frac{15}{2}} - \frac{24}{11}x^{\frac{11}{6}} + C\right) = 4 x^{\frac{13}{2}} - 4 \sqrt[3]{x} \cdot x^{\frac{1}{2}}$$ We can see that our differentiated result matches the original function, confirming our answer is correct.

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Most popular questions from this chapter

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