INDEFINITE INTEGRAL OF RATIONAL FUNCTIONS

Problem 1 :

213x2 - 1 dx

Solution :

Given, 213x2 - 1 dx= 213x-2 - 1 dx=213x-2 dx - 211 dx= 3x-1-1 - x21= -3x - x21= -32 - 2 - -31 - 1= -32 - 2 + 3 +1= -32 - 2 + 4= -32 + 2= -3 + 42= 12

Problem 2 :

-1-2u - 1u2 du

Solution :

Given, -1-2u - 1u2 du= -1-2u - u-2 du=-1-2u du - -1-2u-2 du= u22 - u-1-1-1-2=u22 + 1u-1-2= (-1)22 +1-1 - (-2)22 +1-2= 12 -11 - 42 + 12= 12 - 1 - 2 + 12= 1 - 2 - 4 + 12= -42= -2

Problem 3 :

41 u - 2u du

Solution :

Given, 41 u - 2u du= 41uu - 2u du= 41u - 2u du= 41u12 - 2u-12du= u12 +112 + 1 - 2 × u-12 + 1-12 + 141 = u32 32 - 2 × u121241= (4)32 32 - 2 × (4)121241- (1)32 32 - 2 × (1)121241= (2)332 - 2 × 212 - 132 - 2 × 112= 23 × 8 - 8 - 23 - 4= 163 - 8 - 23 + 4= 143 - 4= 14 - 123= 23

Problem 4 :

812x dx

Solution :

Given, 812x dx= 81 2 (x)-12 dx=2 x-12 + 1-12 + 181=2 x-1 + 22-1 + 2281= 2 x121281= 2 2x1281= 22x81=2 28 - 1= 2 222 - 1= 4 × 2 - 22= 24 - 2

Problem 5 :

10 x - x 3 dx

Solution :

Given, 10 x - x 3 dx= 13 10x - x dx= 13 10x - x12 dx= 13 10x dx - 10x12 dx= 13x2210 - x12 + 112 + 110= 13x2210 - x1 + 22 1 + 2210= 13x2210 - x32 3210= 13x2210 - x32 × 2310= x2610 - x32 × 2910= 126 - 026 - 132 × 29 - 032 × 29= 16 - 29= 3 - 418= -118

Problem 6 :

-1-8 x - x2 2∛x dx

Solution :

Problem 7 :

𝜋40 1 - sin2𝜃cos2𝜃 d𝜃

Solution :

Given, 𝜋40 1 - sin2𝜃cos2𝜃 d𝜃= 𝜋40 cos2 𝜃cos2 𝜃 d𝜃= 𝜋40 1 d𝜃= [𝜃]𝜋40= 𝜋4 - 0= 𝜋4

Problem 8 :

e12x + 1x dx

Solution :

Given, e12x + 1x dx= e12x dx + e1 1x dx= 2x22e1 + xx2e1= x2e1 + 1xe1= e2 - 12 + [In|e| - In|1|]= e2 - 1 + 1 = e2

Problem 9 :

51 x + 1x dx

Solution :

Given, 51 x + 1x dx= 51 xx dx + 51 1x dx= = 51 1 dx + 51 1x dx= [x]51 +In|x|51= [5 - 1] + [In|5| - In|1|]= 4 + [In|5| - In|1|]= 4 + In 5

Problem 10 :

2eecos x - 1x dx

Solution :

Given, 2eecos x - 1x dx= cos x - 1x dx=cos x dx - 1x dx= [sin x - In |x|]2ee= [(sin (2e) - In|2e|) -(sin e - In|e|]= sin (2e) - In|2e| - sin e + In|e|= sin 2e - sin e + In |e||2e|= sin 2e - sin e + In 12= sin 2e - sin e - In 2

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