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Chapter 1: Problem 1

Find an antiderivative of the function: (i) \(f(x)=1-4 x+9 x^{2}\) (ii) \(f(x)=x \sqrt{x}+\sqrt{x}-5\) (iii) \(\mathrm{f}(\mathrm{x})=\sqrt[4]{\mathrm{x}+1}\) (iv) \(\mathrm{f}(\mathrm{x})=(\mathrm{x} / 2-7)^{3}\)

Short Answer

Expert verified
Question: (i) Find the antiderivative of the function \(f(x) = 1 - 4x + 9x^2\). (ii) Find the antiderivative of the function \(f(x) = x\sqrt{x} + \sqrt{x} - 5\). (iii) Find the antiderivative of the function \(f(x) = \sqrt[4]{x + 1}\). (iv) Find the antiderivative of the function \(f(x)=(x / 2-7)^{3}\). Answer: (i) \(F(x) = x - 2x^2 + 3x^3 + C\) (ii) \(F(x) = \frac{2}{5}x^{\frac{5}{2}} + \frac{2}{3}x^{\frac{3}{2}} - 5x + C\) (iii) \(F(x) = \frac{4}{5}(x+1)^{\frac{5}{4}} + C\) (iv) \(F(x) = \frac{1}{8}(x/2 - 7)^4 + C\)

Step by step solution

01

Identify the individual terms in the function

The given function can be broken down into individual terms: \(1\), \(-4x\), and \(9x^2\).
02

Integrate each term individually

Using the power rule of integration, integrate each term (\(ax^n\) becomes \(\frac{1}{n+1}ax^{n+1}\)): $$\int(1)dx = x$$ $$\int(-4x)dx = -2x^2$$ $$\int(9x^2)dx = 3x^3$$
03

Combine the integrated terms and add the constant of integration

Sum the integrated terms and add a constant C: \(F(x) = x - 2x^2 + 3x^3 + C\) #(ii) Find the antiderivative of the function \(f(x) = x\sqrt{x} + \sqrt{x} - 5\)#
04

Rewrite the function using suitable powers

Rewrite the given function as \(f(x) = x^{\frac{3}{2}} + x^{\frac{1}{2}} - 5\).
05

Integrate each term individually

Using power rule of integration, integrate each term: $$\int(x^{\frac{3}{2}})dx = \frac{2}{5}x^{\frac{5}{2}}$$ $$\int(x^{\frac{1}{2}})dx = \frac{2}{3}x^{\frac{3}{2}}$$ $$\int(-5)dx = -5x$$
06

Combine the integrated terms and add the constant of integration

Sum the integrated terms and add a constant C: \(F(x) = \frac{2}{5}x^{\frac{5}{2}} + \frac{2}{3}x^{\frac{3}{2}} - 5x + C\) #(iii) Find the antiderivative of the function \(f(x) = \sqrt[4]{x + 1}\)#
07

Rewrite the function with a suitable power

Rewrite the given function as \(f(x) = (x+1)^{\frac{1}{4}}\).
08

Integrate using the power rule

Using the power rule of integration, integrate the function: $$\int((x+1)^{\frac{1}{4}})dx = \frac{4}{5}(x+1)^{\frac{5}{4}}$$
09

Add the constant of integration

Add a constant C to the integrated function: \(F(x) = \frac{4}{5}(x+1)^{\frac{5}{4}} + C\) #(iv) Find the antiderivative of the function \(f(x)=(x / 2-7)^{3}\)#
10

Identify the function as the composition of two functions

Recognize that the given function is a composition of two functions: \(u(x) = x/2 - 7\) and \(v(u) = u^3\).
11

Use substitution and integrate

Let \(u = x/2 - 7\). Then, \(du = \frac{1}{2}dx\). Rewrite the given function as \(f(x) = v(u(x))du\). Now integrate using substitution: $$\int(u^3\frac{1}{2}du) = \frac{1}{8}u^4$$
12

Replace u with the original expression and add the constant of integration

Replace \(u\) by \(x/2 - 7\) and add a constant C: \(F(x) = \frac{1}{8}(x/2 - 7)^4 + C\)

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Most popular questions from this chapter

Evaluate the following integrals: (i) \(\int \frac{d x}{(1+x)^{3 / 2}+(1+x)^{1 / 2}}\) (ii) \(\int \frac{\mathrm{dx}}{\sqrt[4]{5-x}+\sqrt{5-x}}\) (iii) \(\int \frac{\mathrm{dx}}{\sqrt{(\mathrm{x}+2)}+\sqrt[4]{(\mathrm{x}+2)}}\) (iv) \(\int \frac{\sqrt{x+1}+2}{(x+1)^{2}-\sqrt{x+1}} d x\)

Evaluate \(\int \frac{9 x^{3}-3 x^{2}+2}{\sqrt{3 x^{2}-2 x+1}} d x\)

\(\int\left(x^{3}-2 x^{2}+5\right) e^{3 x} d x\)

Evaluate the following integrals: (i) \(\int \frac{x d x}{\left(x^{2}-3 x+2\right) \sqrt{x^{2}-4 x+3}}\) (ii) \(\int \frac{\left(x^{2}+1\right) d x}{\left(x^{2}+2 x+2\right) \sqrt{(x+1)}}\) (iii) \(\int \frac{(2 x+3) d x}{\left(x^{2}+2 x+3\right) \sqrt{x^{2}+2 x+4}}\)

Evaluate the following integrals : $$ \int \frac{\left(1-x^{2}\right) d x}{x^{1 / 2}\left(1+x^{2}\right)^{3 / 2}} $$
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