$\begin{array}{l}\int\left(-2 x^{3}+4 x-2+\frac{4}{x^{2}}-\frac{3}{\sqrt[3]{x^{2}}}+2 \cdot 3^{x}-\frac{5}{\cos ^{2} x}+\frac{4}{x}-5 \sin x ight) d x= \ =\end{array}$

Question

$\begin{array}{l}\int\left(-2 x^{3}+4 x-2+\frac{4}{x^{2}}-\frac{3}{\sqrt[3]{x^{2}}}+2 \cdot 3^{x}-\frac{5}{\cos ^{2} x}+\frac{4}{x}-5 \sin xight) d x= \ =\end{array}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To integrate the expression, we have: ‖ $$ \int\left(-2 x^{3}+4 x-2+\frac{4}{x^{2}}-\frac{3}{\sqrt[3]{x^{2}}}+2 \cdot 3^{x}-\frac{5}{\cos^{2} x}+\frac{4}{x}-5 \sin x ight) dx = -\frac{x^{4}}{2} - \frac{x^{3}}{\sqrt{3}} + 2 x^{2} - 2 x - \frac{4}{x} + 4 \log(x) + \frac{2 \cdot 3^{x}}{\log(3)} + 5 \cos(x) - 5 an(x) + C $$ ‖ step 2 ⋮ The integral can also be expressed as: ‖ $$ \frac{2 \cdot 3^{x}}{\log(3)} - \frac{3 x^{5} + 2 \sqrt{3} x^{4} - 12 x^{3} + 12 x^{2} - 24 x \log(x) - 30 x \cos(x) + 30 x an(x) + 24}{6 x} + C $$ ‖ step 3 ⋮ Another form of the integral is: ‖ $$ -\frac{4}{x} - 2 x + 2 x^{2} - \frac{x^{3}}{\sqrt{3}} - \frac{x^{4}}{2} + 5 \cos(x) + \frac{2 \cdot 3^{x}}{\log(3)} + 4 \log(x) - \frac{5 \sin(x)}{\cos(x)} + C $$ ‖ step 4 ⋮ The final expression can be simplified further: ‖ $$ -\frac{3 x^{5} \log(3) + 2 \sqrt{3} x^{4} \log(3) - 12 x^{3} \log(3) + 12 x^{2} \log(3) - 4 \cdot 3^{x + 1} x - 24 x \log(3) \log(x) + 30 x \log(3) an(x) - 30 x \log(3) \cos(x) + 24 \log(3)}{6 x \log(3)} + C $$ ∻Answer∻ ⚹ The indefinite integral evaluates to: $$ -\frac{x^{4}}{2} - \frac{x^{3}}{\sqrt{3}} + 2 x^{2} - 2 x - \frac{4}{x} + 4 \log(x) + \frac{2 \cdot 3^{x}}{\log(3)} + 5 \cos(x) - 5 an(x) + C $$ ⚹ ∻Key Concept∻ ⚹ Integration of polynomial and trigonometric functions ⚹ ∻Explanation∻ ⚹ The solution involves integrating a combination of polynomial, logarithmic, and trigonometric functions, applying standard integration techniques. ⚹◊What is the integration by parts formula used to evaluate integrals like $\int x^2 \cdot e^x \, dx$?⍭ Generate me a similar question◊

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