INTEGRATION OF RATIONAL FUNCTIONS

Problem 1 :

8x - 5∛x dx

Solution :

Given, 8x - 5∛x dx= 8x - 5x13 dx= 8xx13 - 5x13 dx= 8x1 - 13 - 5x13 dx= 8x3 - 13 - 5x13 dx= 8x23 - 5x13 dx= 8x23 dx - 5x13 dx= 8 · 3x535 - 5 · 3x232 + C= 24x535 - 15x232 + C

Problem 2 :

2x2 - x + 3x dx

Solution :

Given, 2x2 - x + 3x dxPut x = tx = t212x dx = dtdxx = 2 dt= 2t4 - t2 + 32 · dt = 4t4 - 2t2 + 6dt= 4t4 dt - 2 t2 dt + 6 dt= 4 t55 - 2 t33 + 6t + C= 4 x525 - 2 x323 + 6x12 + C

Problem 3 :

x3 - 1x - 1 dx

Solution :

Given, x3 - 1x - 1 dx= x3 - 13x - 1 dx= (x - 1)x2 + 12 + xx - 1 dx= x2 + 12 + x dx= x2 dx + 1 dx + x dx= x33 + x + x22 + C

Problem 4 :

x3 + 3x2 - 9x - 2x - 2 dx

Solution :

Given, x3 + 3x2 - 9x - 2x - 2 dx

(x3 + 3x2 - 9x - 2) ÷ (x - 2)

Using long division method :

integration-of-rati-fun

Here, the quotient is x2 + 5x + 1.

= x2 + 5x + 1 dx=x2 dx + 5x dx + 1 dx= x33 + 5x22 + x + C

Problem 5 :

t2 + 32t6 dt

Solution :

Given, t2 + 32t6 dt= t22 + 32 + 2t2(3)t6 dt= t4 + 9 + 6t2t6 dt= t4t6 + 9t6 + 6t2t6 dt= 1t2 + 9t6 + 6t4 dt= t-2 + 9t-6 + 6t-4 dt = t-2 dt + 9t-6 dt + 6t-4 dt= t-2 + 1-2 + 1 + 9 · t-6 + 1-6 + 1 + 6 · t-4 + 1-4 + 1 + C= t-1-1 + 9 · t-5-5 + 6 · t-3-3 + C= -t-1 - 9t-55 - 2t-3 + C

Problem 6 :

t + 22t3 dt

Solution :

Given, t + 22t3 dt= t2 + 22 + 2t(2)t3 dt= t + 4 + 4tt3 dt= t + 4 + 4t12t3 dt= tt3 + 4t3 + 4t12t3 dt= 1t2 + 4t3 + 4t12t3 dt= t-2 + 4t-3 + 4t12 - 3 dt = t-2 dt + 4t-3 dt + 4t-52 dt= t-2 + 1-2 + 1 + 4 · t-3 + 1-3 + 1 + 4 · t-52+ 1-52 + 1 + C= t-1-1 + 4 · t-2-2 + 4 · t-32-32 + C= t-1-1 + 4 · t-2-2 + 4-23t-32 + C = t-1-1 + 4 · t-2-2 - 83t-32 + C=-t-1 + 4 -12 t-2 - 83t-32 + C= -t-1 - 2t-2 - 83t-32 + C

Problem 7 :

34 cos u du

Solution :

Given, 34 cos u du= 34 cos u du= 34 sin u + C

Problem 8 :

sec tcos t dt

Solution :

Given, sec tcos t dt= 1cos tcos t dt= 1cos t · 1cos t dt= sec t · sec t dt= sec2t dt= tan t + C

Problem 9 :

1sin2t dt

Solution :

Given, 1sin2t dt= csc2 t dt= -cot t + C

Problem 10 :

sec w sin wcos w dw

Solution :

Given, sec w sin wcos w dw= 1cos w ·sin w cos w dw = sin wcos w cos w dw= sin wcos w · 1cos w dw= sin wcos2 w dwu = cos w, du = -sin w dwdw = 1-sin w du= sin wu · 1u · 1-sin w du= -1u · 1u du=- 1u2 du= -u-2 du= -u-2 + 1-2 + 1=-u-1-1= --u-1= --cos-1 w= sec w + C

Problem 11 :

csc w cos wsin w dw

Solution :

Given, csc w cos wsin w dw= 1sin w ·cos w sin w dw = cos wsin w sin w dw= cos wsin w · 1sin w dw= cos wsin2w dwu = sin w, du = cos w dwdw = 1cos w du= cos wu2 · 1cos w du= 1u2 du= u-2 du= u-2 + 1-2 + 1=u-1-1= -u-1= -csc w + C

Problem 12 :

1 + cot2z cot zcsc z dz

Solution :

Given, 1 + cot2z cot zcsc z dz= csc2z cot zcsc z dz= csc z cot z dz= -csc z + C

Problem 13 :

tan zcos z dz

Solution :

Given, tan zcos z dz= sin zcos zcos z dz= sin zcos z · 1cos z dzu=cos zdu = -sin z dzdz = 1-sin z du = sin zu · 1u · 1-sin z du= -1u · 1u du=-1u2 du= -u-2 du= -u-2 + 1-2 + 1=- u-1-1= --u-1= --cos-1 z= sec z + C

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