92–98. Evaluate the following integrals.
97. ∫ tan⁻¹(∛x) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves the inverse tangent function, tan⁻¹, and a cube root, ∛x. To simplify the integration, consider using substitution. Let u = ∛x, which implies x = u³ and dx = 3u² du.
2. Step 2: Substitute u = ∛x into the integral. Replace tan⁻¹(∛x) with tan⁻¹(u) and dx with 3u² du. The integral becomes ∫ tan⁻¹(u) * 3u² du.
3. Step 3: Factor out the constant 3 from the integral to simplify. The integral now becomes 3 ∫ u² * tan⁻¹(u) du.
4. Step 4: To solve ∫ u² * tan⁻¹(u) du, use integration by parts. Recall the formula for integration by parts: ∫ v dw = uv - ∫ u dv. Let v = tan⁻¹(u) and dw = u² du. Compute dv and u accordingly.
5. Step 5: After applying integration by parts, simplify the resulting expression. Substitute back u = ∛x to express the solution in terms of x.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It can be thought of as the reverse process of differentiation. There are various techniques for integration, including substitution, integration by parts, and recognizing standard forms.

Inverse Functions

Inverse functions are functions that reverse the effect of the original function. For example, the inverse of the tangent function is the arctangent function, denoted as tan⁻¹(x). Understanding how to work with inverse functions is crucial when integrating functions that involve them, as it often simplifies the integration process.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When integrating functions that are composed of other functions, such as tan⁻¹(∛x), recognizing the inner function and applying the chain rule can help in simplifying the integral. This rule is essential for correctly applying substitution methods in integration.