Question

79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
79. ∫ x sin⁻¹(2x) dx

Accepted Answer

Identify the integral to solve: $$\int x \sin$$^{-1}$$(2x) \, dx. $$Notice that it is a product of two functions: $$x$$ and $$\sin$$^{-1}$$(2x)$$, which suggests using integration by parts. Recall the integration by parts formula: $$\int u \, dv = uv - \int v \, du. $$Choose $$u$$ and $$dv$$ wisely. Let $$u = \sin$$^{-1}$$(2x)$$ because its derivative simplifies, and let $$dv = x \, dx. $$Compute $$du by $$differentiating $$u$$: Use the chain rule for $$\sin$$^{-1}$$(2x)$$, so $$du = \frac{2}{\sqrt{1 - (2x)^2$}} \, dx = \frac{2}{\sqrt{1 - 4x^2$}} \, dx. $$Compute $$v by $$integrating $$dv$$: $$v = \int x \, dx = \frac{x^2$}{2}. $$Apply the integration by parts formula: $$\int x \sin$$^{-1}$$(2x) \, dx = u v - \int v \, du = \sin$$^{-1}$$(2x) \cdot \frac{x^2$}{2} - \int \frac{x^2$}{2} \cdot \frac{2}{\sqrt{1 - 4x^2$}} \, dx. $$Simplify the integral: The integral becomes $$\int \frac{x^2$}{\sqrt{1 - 4x^2$}} \, dx. $$This integral may require a substitution or a second table look-up. Consider substituting $$t = 2x or $$using a trigonometric substitution to evaluate it.

79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
79. ∫ x sin⁻¹(2x) dx

Verified video answer for a similar problem:

Key Concepts

Integration by Parts

Inverse Trigonometric Functions

Use of Integral Tables and Computer Algebra Systems

79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.79. ∫ x sin⁻¹(2x) dx

Verified video answer for a similar problem:

Key Concepts

Integration by Parts

Inverse Trigonometric Functions

Use of Integral Tables and Computer Algebra Systems

79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
79. ∫ x sin⁻¹(2x) dx