Answered: f' (x) h' (x) 1)(x³ 3x) — (а? — 1)(23 —…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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u-Substitution

Find the antiderivatives using the technique of U-Substitution

f' (x) = (x² – 1)(æ³ – 3x)80
h' (x) = x cos(2 – 3x?)

Expert Solution

Step 1: Consider the provided derivative,

$f' (x) = (x^{2} - 1) {(x^{3} - 3 x)}^{90}$

Integrate the above derivative.

$\begin{array}{rcl} f (x) & = & \int (x^{2} - 1) {(x^{3} - 3 x)}^{90} d x \\ T a k e x^{3} - 3 x & = & t \Rightarrow (x^{2} - 1) d x = \frac{1}{3} d t \\ f (x) & = & \frac{1}{3} \int {(t)}^{90} d t \\ = & \frac{1}{3} [\frac{t^{90 + 1}}{90 + 1}] + C \\ = & \frac{1}{3} [\frac{t^{91}}{91}] + C \\ = & \frac{1}{273} {(x^{3} - 3 x)}^{91} + C \end{array}$

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