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))² 

(6)  ∫(1 - x)³ 

(7)  (1/x⁵)

(8)  1/x^5/2

(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/x^5/2

Solution :

∫ x^(-5/2) dx  =  x^(-5/2)+1]/(-5/2)+1)

=  x^(-3/2)/(-3/2)+C

=  (-2/3)x^(-3/2)+C

= -2/3x^3/2+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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