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))2
(6) ∫(1 - x)3
(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)/x2 dx
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))2 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)3 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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