Determine the following indefinite integral. Check your work by differentiation.

Ask by Gordon Smith. in the United States

Jan 31,2025

Question

Determine the following indefinite integral. Check your work by differentiation. $$ \int\left(5 x^{\frac{2}{3}}+4 x^{-\frac{2}{3}}+7\right) d x $$

UpStudy AI · Accepted Answer

Calculate the integral $$ \int(5x^{2/3}+4x^{-2/3}+7)dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int \left(5x^{\frac{2}{3}}+4x^{\frac{-2}{3}}+7\right) dx$$
- step1: Rewrite the fraction:
  $$\int \left(5x^{\frac{2}{3}}+4x^{-\frac{2}{3}}+7\right) dx$$
- step2: Use properties of integrals:
  $$\int 5x^{\frac{2}{3}} dx+\int 4x^{-\frac{2}{3}} dx+\int 7 dx$$
- step3: Evaluate the integral:
  $$3x^{\frac{5}{3}}+\int 4x^{-\frac{2}{3}} dx+\int 7 dx$$
- step4: Evaluate the integral:
  $$3x^{\frac{5}{3}}+12x^{\frac{1}{3}}+\int 7 dx$$
- step5: Evaluate the integral:
  $$3x^{\frac{5}{3}}+12x^{\frac{1}{3}}+7x$$
- step6: Simplify:
  $$3x\sqrt[3]{x^{2}}+12x^{\frac{1}{3}}+7x$$
- step7: Simplify:
  $$3x\sqrt[3]{x^{2}}+12\sqrt[3]{x}+7x$$
- step8: Add the constant of integral C:
  $$3x\sqrt[3]{x^{2}}+12\sqrt[3]{x}+7x + C, C \in \mathbb{R}$$
It seems there was an issue with the differentiation. Let's try to differentiate the result manually.

The derivative of $$3x\sqrt[3]{x^{2}}+12\sqrt[3]{x}+7x + C$$ with respect to $$x$$ is:

$$
\frac{d}{dx}(3x\sqrt[3]{x^{2}}+12\sqrt[3]{x}+7x + C) = 5x^{\frac{2}{3}}+4x^{-\frac{2}{3}}+7
$$

This matches the original function, confirming that the integration was correct.
The indefinite integral is $$3x\sqrt[3]{x^{2}}+12\sqrt[3]{x}+7x + C$$. Differentiating this result gives back the original function, confirming the integration is correct.

Determine the following indefinite integral. Check your work by differentiation.

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