PRACTICE QUESTIONS ON INTEGRATION

Integrate the following with respect to x

(1)  3x3+7x2-2x+1

(2)  (7x2+2x+3)/x5

(3)  (x-1) (x+2)

(4)  (x + 1)/x2

(5)  (x + (1/x))

(6)  (1 - x)

(7)  (1/x5)

(8)  1/x5/2

(9)  1/∛x⁵ 

(10)  (1/x3)(1/4)

Question 1 :

∫(3x3+7x2-2x+1) dx

Solution :

=  ∫(3x3+7x2-2x+1) dx

Now we are going to write separate integration.

=  ∫3x3 dx+∫7x2dx-∫2x dx+∫1 dx

=  3∫x3 dx+7∫x2dx-2∫xdx+∫1dx

=  3(x4/4)+7 (x3/3)-2(x2/2)+x+C

=  3x4/4 + 7x3/3 - x2 + x + C

So, the answer is 3x4/4 + 7x3/3 - x2 + x + C.

Question 2 :

∫(7x2+2x+3)/x5 dx

Solution :

=  ∫[(7x2+2x+3)/x5] dx

Decomposing it, we get

=  ∫(7x2/x5) dx + ∫(2x/x5) dx + ∫(3/x⁵) dx

=  7∫x-3 dx + 2∫x-4 dx + 3∫x-5 dx

=  (7x-2)/(-2) + (2x-3)/(-3) + 3x-4/(-4) + C

=  -7/2 x2 - 2/3x3 - 3/4x4 + C

So, the answer is -7/2 x2 - 2/3x3 - 3/4x4 + C.

Question 3 :

∫(x-1) (x+2) dx

Solution :

=  ∫ (x-1) (x+2) dx

=  ∫(x2+x-2) dx

=  ∫ x2 dx + ∫x dx - ∫2 dx

=  (x3/3) + (x2/2) - 2x + C

So, the answer is (x3/3) + (x2/2) - 2x + C.

Question 4 :

∫(x + 1)/xdx

Solution :

=  ∫(x+1)/x2 dx

=  ∫(x/x2) dx+∫(1/x2) dx

=  ∫(1/x) dx + ∫(x⁻²) dx

=  log x - x-1

=  log x - 1/x + C

So, the answer is log x - 1/x + C.

Question 5 :

∫(x + (1/x))dx

Solution :

(x + (1/x))2  =  x2 + (1/x2) + (2  x ⋅ 1/x )

=  ∫x2 dx+∫(1/x2) dx+2∫dx

=  ∫x2 dx + ∫x-2 dx + 2∫dx

=  (x3/3) - 1/x + 2x + C

So, the answer is (x3/3) - 1/x + 2x + C.

Question 6 :

∫(1 - x)dx

Solution :

(1 - x)3  =  13-3(1)2x+3x2-x3

=  1-3x+3x2-x3

∫(1-x)3 dx  =  ∫(1-3x+3x2-x3) dx

=  ∫1 dx - ∫3x dx + ∫3x2 dx - ∫x3 dx

=  x-3x2/2 + 3x3/3-x4/4 + C

=  x - 3x2/2+x3-x4/4 + C

So, the answer is x - 3x2/2+x3-x4/4 + C.

Question 7 :

∫ (1/x5) dx

Solution :

1/x5  =  x-5

=  x-4/(-4) + C

=  -1/4x4 + C

So, the answer is -1/4x4 + C.

Question 8 :

1/x5/2

Solution :

∫ x(-5/2) dx  =  x(-5/2)+1]/(-5/2)+1)

=  x(-3/2)/(-3/2)+C

=  (-2/3)x(-3/2)+C

= -2/3x3/2+C

=  -2/3x√x + C

So, the answer is -2/3x√x + C.

Question 9 :

∫ 1/∛x⁵ dx

Solution :

1/∛x⁵  =  x(-5/3)

∫ x(-5/3) dx  = x(-5/3)+1)/(-5/3)+1

=  x(-2/3)/(-2/3) + C

=  -3/2x(2/3) + C

So, the answer is -3/2x(2/3) + C.

Question 10 :

(1/x3)(1/4)

Solution :

(1/x3)(1/4)  =  x(-3/4)

∫ x(-3/4) dx  =  x(-3/4)+1)/(-3/4)+1)

=  x(1/4)/(1/4) + C

=  4x(1/4) + C

So, the answer is 4x(1/4) + C.

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