Integral of x^3 / (x^2 + 4) * tan^(-1)(x/2) dx

Math Problem Statement

∫ (x^3 / (x^2 + 4)) * tan^(-1)(x/2) dx

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From the provided image, the equation seems to be:

$\int \frac{x^3}{x^2 + 4} \cdot \text{tan}^{-1}\left(\frac{x}{2}\right) dx$

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Here are 5 related questions:

What is the general method for solving integrals involving inverse trigonometric functions?
How do you approach integrals with products of algebraic and trigonometric functions?
What are the common techniques used for integration by parts?
How does substitution work in simplifying integrals?
Can integrals involving tangent inverse functions always be solved analytically?

Tip: When dealing with complex integrals, breaking the expression into simpler parts can make the solution easier.

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Integration
Inverse Trigonometric Functions

Integration by Parts: ∫ u dv = uv - ∫ v du
Inverse Tangent Function: tan^(-1)(x)

Integration by Parts
Substitution Method

Undergraduate Level (Calculus II)

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