9.1.6 Thedu=f′.x/dxsubstitution ➤263 282➤➤
Integrate the following functions by means of an appropriate substitution.
(i)x+ 1
x^2 + 2 x+ 3(ii) xsin(x^2 + 1 ) (iii) cosxesinx (iv) sinxcosx- Compare the results of (iv) with that of Q9.1.8(iv). Are the answers the same? Explain.
9.1.7 Integrating rational functions ➤265 283➤➤
A.Find
∫
dx
x^2 +x− 2using partial fractions.B. Find
∫
dx
x^2 + 2 x+ 2, given that∫
dx
x^2 + 1=tan−^1 x.C.Find
∫
2 x^2 + 5 x+ 4
x^2 + 2 x+ 2dxHint: divide out first and think about Q9.1.6 andBimmediately above.D.Integrate
(i)1
x^2 +x+ 1(ii)1
x^2 + 3 x+ 2(iii)2 x+ 1
x^2 +x− 1(iv)3 x
(x− 1 )(x+ 1 )9.1.8 Using trig identities in integration ➤269 283➤➤
Integrate the following functions using appropriate trig identities.
(i) sin^2 x (ii) cos^3 x (iii) sin 2xcos 3x
(iv) sinxcosx- Cf (iv) with Q9.1.6(iv) – are the answers the same?
9.1.9 Using trig substitutions in integration ➤272 283➤➤
Integrate
(i)1
√
9 −x^2(ii)1
√
3 − 2 x−x^29.1.10 Integration by parts ➤273 283➤➤
Integrate by parts
(i) xsinx (ii) x^2 ex (iii) exsinx