∫(√(x))/(√(1+x^3)) d x

Name: Problem 80, Chapter 5: Calculus of a Single Variable
Uploaded: 2021-03-02T17:18:14-08:00
Duration: 3 min 42 s
Channel: Anurag Kumar
Description: ∫(√(x))/(√(1+x^3)) d x

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Key Concepts

u-substitution

u-substitution is an integration technique where a new variable is introduced to simplify the integral by taking advantage of a function and its derivative present in the integrand. This method transforms a complicated expression into a simpler one by substituting u = g(x) and rewriting the dx in terms of du, eventually leading to an easier integral to evaluate.

Recognition of Standard Integral Forms

Recognizing standard integral forms involves identifying parts of the integrand that match known integral formulas. This might include forms such as ? du/?(1+u²), which is directly integrable by using inverse hyperbolic functions. Recognizing these forms expedites the integration process by allowing the use of established antiderivatives.

Inverse Hyperbolic Functions

The inverse hyperbolic functions, such as sinh?¹(u), arise in integrals that involve expressions like 1/?(1+u²). These functions have standard antiderivatives that can be used to solve integrals, thereby simplifying the process of integration when such patterns appear. Understanding their properties helps in directly evaluating integrals that match these classical forms.

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