Integration For A Level Maths
Integration For A Level Maths – Introduction
In this article we are going to continue looking at a number of questions that involve integration for A Level maths.
Integration for A Level maths is a huge topic and it is one that cannot be granted as it is found in pure maths and also mechanics.
The whole topic of calculus which covers both differentiation and integration is a huge area and can encompass a number of topic areas. You may be asked to do a polynomial division and then find the area under a particular part of the polynomial and integration also is found within Parametric Equations.
In this article the topic of integration for A Level maths will be looked at from the point of view of a student who is in year 12. You will have covered the topic of differentiation and the next logical topic that follows is that of integration.
Make sure that you are fully familiar with all the techniques for differentiation and indices as these skills will be needed with the longer questions that are based on integration.
There are 15 questions and 15 detailed solutions on how to solve the questions related to integration for A Level maths. You will see a range of questions that involve differential equations, definite integrals as well as indefinite integration.
Integration For A Level Maths - Questions 1 - 5
Q1.
Solution
\begin{aligned} \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\left(x^2+3\right)^2}{x^2} & =\frac{x^4+6 x^2+9}{x^2} \\ & =x^2+6+9 x^{-2} . \end{aligned}
Solution
\begin{aligned} \frac{\mathrm{d} y}{\mathrm{~d} x}=x^2+6+9 x^{-2} & \Rightarrow y=\frac{1}{3} x^3+6 x+9\left(\frac{x^{-1}}{-1}\right)+c \\ & \Rightarrow y=\frac{1}{3} x^3+6 x-9 x^{-1}+c . \end{aligned}Passes (3, 20)
\begin{aligned} &20=9+18-3+c \Rightarrow c=-4\\ &y=\frac{1}{3} x^3+6 x-9 x^{-1}-4 . \end{aligned}Q2.
Solution
\begin{aligned} \int\left(12 x^5-8 x^3+3\right) \mathrm{d} x & =12 \times \frac{1}{6} x^6-8 \times \frac{1}{4} x^4+3 x+c \\ & =2 x^6-2 x^4+3 x+c . \end{aligned}Q3.
Passes (4, 22)
\begin{aligned} 22 & =64-16-28+c \Rightarrow c=2 \\ \mathrm{f}(x) & =x^3-2 x^{\frac{3}{2}}-7 x+2 . \end{aligned}Q4.
Solution
\begin{aligned} \int\left(2 x^3+\frac{3}{x^2}\right) \mathrm{d} x & =\int\left(2 x^3+3 x^{-2}\right) \mathrm{d} x \\ & =2 \times \frac{1}{4} x^4+3 \times\left(\frac{x^{-1}}{-1}\right)+c \\ & =\frac{1}{2} x^4-3 x^{-1}+c . \end{aligned}Q5.
Solution
\begin{aligned} \frac{\mathrm{d} y}{\mathrm{~d} x}=5 x^{-\frac{1}{2}}+x \sqrt{x} & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=5 x^{-\frac{1}{2}}+x^{\frac{3}{2}} \\ & \Rightarrow y=5 \times 2 x^{\frac{1}{2}}+\frac{2}{5} x^{\frac{5}{2}}+c \\ & \Rightarrow y=10 x^{\frac{1}{2}}+\frac{2}{5} x^{\frac{5}{2}}+c . \end{aligned}Passes (4, 35)
\begin{aligned} &35=20+12.8+c \Rightarrow c=2.2\\ &y=10 x^{\frac{1}{2}}+\frac{2}{5} x^{\frac{5}{2}}+2.2 . \end{aligned}Integration For A Level Maths - Questions 6 - 10
Q6.
Solution
\begin{aligned} \int\left(8 x^3+6 x^{\frac{1}{2}}-5\right) \mathrm{d} x & =8 \times \frac{1}{4} x^4+6 \times \frac{2}{3} x^{\frac{3}{2}}-5 x+c \\ & =2 x^4+4 x^{\frac{3}{2}}-5 x+c . \end{aligned}Q7.
Passes (4, 5)
\begin{aligned} &5=24-20-8+c \Rightarrow c=9\\ &y=\frac{3}{2} x^2-10 x^{\frac{1}{2}}-2 x+9 \end{aligned}Q8.
Solution
\begin{aligned} \int\left(12 x^5-3 x^2+4 x^{\frac{1}{3}}\right) \mathrm{d} x & =12 \times \frac{1}{6} x^6-3 \times \frac{1}{3} x^3+4 \times \frac{3}{4} x^{\frac{4}{3}}+c \\ & =2 x^6-x^3+3 x^{\frac{4}{3}}+c . \end{aligned}Passes (-1, 0)
\begin{aligned} &0=-4-4-1+c \Rightarrow c=9\\ &\mathrm{f}(x)=4 x^3-4 x^2+x+9 . \end{aligned}Q9.
Solution
\begin{aligned} \mathrm{f}^{\prime}(x)=12 x^2-8 x+1 & \Rightarrow \mathrm{f}(x)=12 \times \frac{1}{3} x^3-8 \times \frac{1}{2} x^2+x+c \\ & \Rightarrow \mathrm{f}(x)=4 x^3-4 x^2+x+c . \end{aligned}Passes (-1, 0)
\begin{aligned} &0=-4-4-1+c \Rightarrow c=9\\ &\mathrm{f}(x)=4 x^3-4 x^2+x+9 . \end{aligned}Q10.
Solution
\begin{aligned} \int\left(2 x^5+7+\frac{1}{x^3}\right) \mathrm{d} x & =\int\left(2 x^5+7+x^{-3}\right) \mathrm{d} x \\ & =2 \times \frac{1}{6} x^6+7 x+\frac{x^{-2}}{-2}+c \\ & =\frac{1}{3} x^6+7 x-\frac{1}{2} x^{-2}+c . \end{aligned}
Integration For A Level Maths – Questions 11 – 15
Solution
\frac{6 x+3 x^{\frac{5}{2}}}{\sqrt{x}}=\frac{6 x+3 x^{\frac{5}{2}}}{x^{\frac{1}{2}}}=\underline{\underline{6 x^{\frac{1}{2}}+3 x^2}}
Solution
\begin{aligned} \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6 x+3 x^{\frac{5}{2}}}{\sqrt{x}} & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=6 x^{\frac{1}{2}}+3 x^2 \\ & \Rightarrow y=6 \times \frac{2}{3} x^{\frac{3}{2}}+3 \times \frac{1}{3} x^3+c \\ & \Rightarrow y=4 x^{\frac{3}{2}}+x^3+c . \end{aligned}Passes (4, 90)
\begin{aligned} &90=32+64+c \Rightarrow c=-6\\ &y=4 x^{\frac{3}{2}}+x^3-6 . \end{aligned}Q12.
Solution
\begin{aligned} \int\left(x^4+6 x^{\frac{1}{2}}\right) \mathrm{d} x & =\frac{1}{5} x^5+6 \times \frac{2}{3} x^{\frac{3}{2}}+c \\ & =\frac{1}{5} x^5+4 x^{\frac{3}{2}}+c . \end{aligned}Q13.
Solution
\begin{aligned} \mathrm{f}^{\prime}(x)=3 x^2-3 x+5 & \Rightarrow \mathrm{f}(x)=3 \times \frac{1}{3} x^3-3 \times \frac{1}{2} x^2+5 x+c \\ & \Rightarrow \mathrm{f}(x)=x^3-\frac{3}{2} x^2+5 x+c . \end{aligned}Passes (2, 10)
\begin{aligned} &10=8-6+10+c \Rightarrow c=-2\\ &f(x)=x^3-\frac{3}{2} x^2+5 x-2 .\\ &f(1)=1-\frac{3}{2}+5-2=2 \frac{1}{2} . \end{aligned}Q14.
Solution
\begin{aligned} \int\left(6 x^2+\frac{2}{x^2}+5\right) \mathrm{d} x & =\int\left(6 x^2+2 x^{-2}+5\right) \mathrm{d} x \\ & =6 \times \frac{1}{3} x^3+2\left(\frac{x^{-1}}{-1}\right)+5 x+c \\ & =2 x^3-2 x^{-1}+5 x+c . \end{aligned}Q15.
Passes (4, -1)
\begin{aligned} &-1=4-24+12+c \Rightarrow c=7\\ &f(x)=\frac{1}{4} x^2-12 x^{\frac{1}{2}}+3 x+7 \end{aligned}Most of these questions are covering either solving a differential equation or having to deal with an indefinite integral. With indefinite integrals it is always important to remember the constant “c”. This alone is worth one mark.
Integration for A Level Maths covers a lot of essential skills in maths and you will have seen from the above that the use of indices is widely used throughout. It is important that you are able to rewrite any expressions in a form which can be successfully integrated.