MYTUTOR SUBJECT ANSWERS

163 views

How do I solve an integration by substitution problem?

I think it’s best I work through an example with you as these problems can vary quite a lot, but the general methods used are the same.

Example: Use the substitution u=x2+5 to find: The integral of (x3/sqrt(x2+5)).dx between the limits of 2 and 1.

So the whole idea of using a substitution here is to simplify the integration for us. The first thing we must do is substitute the given substitution in, otherwise there wouldn’t be much point! In doing this, we also need to replace the .dx in the integration, we do this by finding du/dx and then rearranging for dx; in this particular example, du/dx = 2x, so dx = (1/2x)du, so the integral becomes (x3/sqrt(x2+5)). (1/2x).du =( x2/2sqrt(x2+5)).du. We then use the substitution to get the x’s in terms of u: the numerator, x2 becomes u-5 (as u=x2+5), the denominator, 2sqrt(x2+5) becomes 2sqrt(u). Finally, modifying the limits in terms of u: since the top limit is x=2, then this is equivalent to u=22+5=9, and the bottom limit becomes 12+5=6.

We now have our rephrased integration problem: Integrate ((u-5)/2sqrt(u)).du between the limits of 9 and 6. Notice we can split up the integrand (the thing we’re integrating): =(u/2sqrt(u))-(5/2sqrt(u)). You know that this is equivalent to 0.5u1/2-5.5u-1/2, which when integrated is (1/3)u3/2-5u1/2. The only thing left to do now is apply the limits: [(1/3)(93/2)-5(91/2)]-[ [(1/3)(63/2)-5(61/2)] = 9-15-2sqrt(6)+5sqrt(6) = 7sqrt(6)-6

The steps to solving other substitution problems are very similar to the ones detailed above, obviously the manipulations will be different, but the ideas are the same.

Andrew D. A Level Maths tutor, GCSE Maths tutor, A Level Further Math...

5 months ago

Answered by Andrew, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

244 SUBJECT SPECIALISTS

£20 /hr

Ruadhan P.

Degree: Physics (Masters) - Nottingham University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Chemistry
-Personal Statements-

“I am a currently studying Physics at the University of Nottingham. The way Physics and Maths are able to explain the Universe around us truly fascinates me and I hope to be able to share that fascination and excitement for the subject...”

£20 /hr

Jack A.

Degree: Physics (Masters) - Bristol University

Subjects offered: Maths, Physics

Maths
Physics

“Hi, I'm Jack, University of Bristol Physics student, and two years experienced Maths GCSE tutor. I have always found myself excited in science, the core of my joy in learning and teaching Maths and Physics comes from my strong desire t...”

MyTutor guarantee

£20 /hr

Hannah J.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered: Maths, Science+ 3 more

Maths
Science
Physics
Further Mathematics
French

“I'm a first year Maths student at Durham University and I've always loved Science and Maths. Having taught swimming for 2 years, and English, French and Maths in a primary school for a year, I have become a patient and understanding t...”

MyTutor guarantee

About the author

£20 /hr

Andrew D.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: Maths, Further Mathematics

Maths
Further Mathematics

“About me Hi, I'm Andrew. I'm currently going into my second year studying maths at the University of Warwick. I find maths incredibly fascinating, as my passion stems from the idea thatmaths is everywhere and is fundamental to underst...”

You may also like...

Posts by Andrew

How do I find the limit as x-->infinity of (4x^2+5)/(x^2-6)?

How do I rationalise the denominator of a fraction?

How do I solve an integration by substitution problem?

How do I use proof by induction?

Other A Level Maths questions

Differentiate: y = xsin(x)

How will you simplify (3 xsquare root of 2) to the square?

What is the equation of the normal line to the curve y = 3x^3 - 6x^2 at the point (1, 4)?

∫ log(x) dx

View A Level Maths tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok