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...

10 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


£26 /hr

Scott R.

Degree: PGCE Secondary Mathematics (Other) - Leeds University

Subjects offered:Maths, Further Mathematics

Further Mathematics

“I am currently completing 2 PGCEs in Leeds. I have always had a passion for maths and my objective is to help as many as possible reach their full potential.”

£20 /hr

Praveenaa K.

Degree: Mathematics (Masters) - Bristol University

Subjects offered:Maths, Science+ 4 more

Further Mathematics
-Personal Statements-

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

£20 /hr

Luke B.

Degree: Mathematics (Masters) - Sheffield University

Subjects offered:Maths, Further Mathematics + 3 more

Further Mathematics
-Personal Statements-

“I am a fun, engaging and qualified tutor. I'd love to help you with whatever you need, giving you the support you need to be the best you can be!”

About the author

Andrew D. A Level Maths tutor, GCSE Maths tutor, A Level Further Math...
£20 /hr

Andrew D.

Degree: Mathematics (Masters) - Warwick University

Subjects offered:Maths, Further Mathematics

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...”

MyTutor guarantee

|  1 completed tutorial

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

Find the gradient of y=6x^3+2x^2 at (1,1)

Solve the following simultaneous equations: 3x + 5y = -4 and -2x + 3y = 9

Differentiate with respect to x: y=xln(x)

What is the difference between differentiation and integration, and why do we need Calculus at all?

View A Level Maths tutors

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