Understanding Engineering Mathematics

Solution to review question 9.1.9

Whenever you see something like

√

a^2 −x^2 try a sine substitution

x=asinθ.

(i) For

∫

dx

√

9 −x^2

the function

√

9 −x^2 =

√

32 −x^2 appears and so we

try a substitutionx=3sinθ.Thisgivesdx=3cosθdθand

√

9 −x^2 =

√

9 −9sin^2 θ

= 3

√

cos^2 θ=3cosθ

So ∫

dx

√

9 −x^2

−−−→

∫

3cosθdθ

3cosθ

=

∫

dθ=θ+C

But sinθ=

x

3

,soθ=sin−^1

(x

3

)

and therefore

∫

dx

√

9 −x^2

=sin−^1

(x

3

)

+C

(ii)

∫

dx

√

3 − 2 x−x^2

doesn’t seem to fit any of the simple forms given

above. However, by completing the square and substituting we can

make progress:

∫

dx

√

3 − 2 x−x^2

≡

∫

dx

√

4 −(x+ 1 )^2

≡

∫

dx

√

22 −(x+ 1 )^2

=sin−^1

(

x+ 1

2

)

+C

on puttingu=x+1.

9.2.10 Integration by parts

➤

252 283➤

Useful for some products the integration by parts formula is derived by integrating the
product rule of differentiation:

d(uv)

dx

=u

dv

dx

+v

du

dx

∫

u

dv

dx

dx=uv−

∫

v

du

dx

dx

(‘integrate one, differentiate the other’).