Math
ski
Worksheet
Questions
Solutions
Polynomial Integration I
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Name
Question 1
Integrate the following:
(a)
\(4x^3\)
(b)
\(6x^2-3x+2\)
(c)
\(\sqrt{x}\)
(d)
\(\dfrac{1}{x^3}\)
(e)
\(\dfrac{3}{\sqrt{x}}\)
(f)
\(x^{3/2}-x^{-1/2}\)
(a)
Answer
\(x^4+C\)
(b)
Answer
\(2x^3- frac{3}{2}x^2+2x+C\)
(c)
Rewrite as \(x^{1/2}\)
Answer
\( frac{2}{3}x^{3/2}+C\)
(d)
Rewrite as \(x^{-3}\)
Answer
\(- frac{1}{2}x^{-2}+C\)
(e)
Rewrite as \(3x^{-1/2}\)
Answer
\(6\sqrt{x}+C\)
(f)
Answer
\( frac{2}{5}x^{5/2}-2x^{1/2}+C\)
Question 2
By first simplifying, integrate the following:
(a)
\((x+2)(x-3)\)
(b)
\(\dfrac{x^3+4x}{x}\)
(c)
\((2x+1)^2\)
(a)
Expand
\(x^2-x-6\)
Answer
\( frac{x^3}{3}- frac{x^2}{2}-6x+C\)
(b)
Simplify
\(x^2+4\)
Answer
\( frac{x^3}{3}+4x+C\)
(c)
Expand
\(4x^2+4x+1\)
Answer
\( frac{4x^3}{3}+2x^2+x+C\)
Question 3
Evaluate the following definite integrals:
(a)
\(\displaystyle\int_1^3 (2x+1)\,dx\)
(b)
\(\displaystyle\int_0^4 \sqrt{x}\,dx\)
(c)
\(\displaystyle\int_1^2 \dfrac{3}{x^2}\,dx\)
(a)
Integrate then substitute
\([x^2+x]_1^3=(9+3)-(1+1)\)
Answer
\(10\)
(b)
Integrate
\([ frac{2}{3}x^{3/2}]_0^4= frac{2}{3}\cdot8\)
Answer
\( frac{16}{3}\)
(c)
Integrate
\([-3x^{-1}]_1^2=(- frac{3}{2})-(-3)\)
Answer
\( frac{3}{2}\)
Question 4
Find \(f(x)\) given \(f'(x)=3x^2-4x+1\) and \(f(2)=5\).
Integrate
\(f(x)=x^3-2x^2+x+C\)
Use \(f(2)=5\)
\(8-8+2+C=5\Rightarrow C=3\)
Answer
\(f(x)=x^3-2x^2+x+3\)
Question 5
Find the area enclosed between \(y=x^2-1\) and the \(x\)-axis.
Roots
\(x=\pm1\)
Area (curve below axis)
\(\displaystyle\left|\int_{-1}^1(x^2-1)\,dx ight|=\left|[ frac{x^3}{3}-x]_{-1}^1 ight|=\left|( frac{1}{3}-1)-(- frac{1}{3}+1) ight|=\left|- frac{4}{3} ight|\)
Answer
\(\dfrac{4}{3}\) square units
Generated by Mathski · mathski.io · IB Mathematics AA Shadow Worksheets