Evaluate the definite integral. ∫0^1 √(2 x)(√(x)+√(2)) d x |...

Name: Problem 38, Chapter 6: Applied Calculus for the Managerial, Life, and Social Sciences
Uploaded: 2021-05-17T16:59:44-08:00
Duration: 4 min 5 s
Channel: Anurag Kumar
Description: Evaluate the definite integral. ∫0^1 √(2 x)(√(x)+√(2)) d x

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Key Concepts

Power Rule for Integration

The power rule is a basic method for integrating functions of the form x^n, where n is any real number except -1. This rule states that the integral of x^n is x^(n+1)/(n+1) plus a constant of integration (for indefinite integrals), and it is particularly useful after rewriting radical expressions in exponent form.

Definite Integral

A definite integral represents the accumulation of quantities over a specific interval and is often interpreted as the area under a curve. It is computed by finding an antiderivative of the function and evaluating it at the endpoints of the interval, subtracting the value at the lower limit from that at the upper limit.

Radical Functions

Radical functions, such as square roots, are expressions that involve roots of variables. When integrating these, it is common to rewrite them in exponential form (for example, ?x as x^(1/2)) to facilitate the use of standard integration techniques.

Linearity of Integration

The linearity property of integrals allows one to split the integral of a sum into the sum of individual integrals and to factor out constant multipliers. This property simplifies the evaluation of integrals by enabling term-by-term integration and simplification before applying other integration techniques.

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