Verify the integration formulas in Exercises 111–114.
113. ∫ (arcsin x)² dx = x(arcsin x)² - 2x + 2 √(1 - x²) arcsin x + C

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Identify the integral to verify: \(\int (\arcsin x)^2 \, dx\).

Use integration by parts, letting \(u = (\arcsin x)^2\) and \(dv = dx\). Then compute \(du\) and \(v\): - \(du = 2 \arcsin x \cdot \frac{1}{\sqrt{1 - x^2}} \, dx\) (using the derivative of \(\arcsin x\)), - \(v = x\).

Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), which gives \(\int (\arcsin x)^2 \, dx = x (\arcsin x)^2 - 2 \int \frac{x \arcsin x}{\sqrt{1 - x^2}} \, dx\).

To evaluate \(\int \frac{x \arcsin x}{\sqrt{1 - x^2}} \, dx\), use substitution: let \(t = \arcsin x\), so \(x = \sin t\) and \(dx = \cos t \, dt\). Rewrite the integral in terms of \(t\) and simplify.

After substitution, integrate the resulting expression with respect to \(t\), then substitute back \(t = \arcsin x\) to express the answer in terms of \(x\). Combine all parts to verify the given formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. This method is essential for integrating products involving inverse trigonometric functions like arcsin.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin(x), are the inverses of the standard trigonometric functions. Understanding their derivatives and properties is crucial, as these functions often appear inside integrals and require special techniques for integration.

Algebraic Manipulation and Simplification

After applying integration techniques, simplifying the resulting expressions using algebraic manipulation is necessary. This includes factoring, combining like terms, and using identities like 1 - x² under square roots to match the given formula and verify the integral.

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