Question

Evaluate the integral. $$\int_{0}^{1}\left(x^{3 / 4}-2 x^{1 / 2}\right) d x$$

Name: Problem 14, Chapter 5: Calculus: One Variable
Uploaded: 2021-07-28T14:59:51-08:00
Duration: 1 min 26 s
Channel: Matthys Marthinus
Description: Evaluate the integral. ∫0^1(x^3 / 4-2 x^1 / 2) d x

   Evaluate the integral.
$$\int_{0}^{1}\left(x^{3 / 4}-2 x^{1 / 2}\right) d x$$

Calculus: One Variable

Garret J. Etgen,… 10th Edition

Chapter 5, Problem 14

Instant Answer

Solved by Expert Matthys Marthinus

07/28/2021

Step 1

The antiderivative of $x^{3/4}$ is $\frac{4}{7}x^{7/4}$ and the antiderivative of $-2x^{1/2}$ is $-\frac{4}{3}x^{3/2}$. So, the antiderivative of the function is $\frac{4}{7}x^{7/4}-\frac{4}{3}x^{3/2}$. Show more…

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Evaluate the integral. $$\int_{0}^{1}\left(x^{3 / 4}-2 x^{1 / 2}\right) d x$$

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Key Concepts

Algebraic Simplification of Fractional Exponents

Integrals involving fractional exponents require careful algebraic manipulation. Working with exponents like 3/4 or 1/2 involves converting expressions into equivalent forms to simplify and correctly apply the integration techniques.

Definite Integration

Definite integrals compute the accumulated value, such as area under a curve, between two specific limits. The fundamental theorem of calculus connects antiderivatives with definite integrals, enabling the evaluation of an integral from a lower to an upper bound by taking the difference of the antiderivative evaluated at these bounds.

Power Rule for Integration

This rule is essential for integrating functions of the form x^n, where n is any real number except -1. It states that the integral of x^n with respect to x is x^(n+1)/(n+1) + C. This formula allows one to compute antiderivatives for power functions easily, thereby simplifying the integration process.

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