INTEGRATION WORKSHEET WITH SOLUTONS

Integrate the following with respect to x

(1)  5x⁴+3(2x+3)⁴-6(4-3x)⁵

(2)  (3/x)+[m/(4x+1)]-2(5-2x)⁵

(3)  4 - 5/(x+2) + 3 cos 2x

(4)  3e^7x-4sec(4x+3)tan(4x+3)+11/x⁵

(5)  pcosec²(px-q)-6(1-x)⁴+4e^(3-4x)

(6)  (2x-5) (36+4x)

(7)  (1 + x³)²

(8)  (x³+4x²-3x+2)/x²

(9)  (x⁴ - x² + 2)/(x + 1)

(10)  (1+x)²/√x

Question 1 :

 5x⁴+3(2x+3)⁴-6(4-3x)⁵

Solution:

Now we are going to integrate the given function

=  ∫[5x⁴+3(2x+3)⁴-6(4-3x)⁵] dx

=  ∫5x⁴ dx + 3∫(2x+3)⁴  dx - 6∫(4-3x)⁵ dx

=  x⁵ dx + (3/10)(2x+3)⁵ + (1/3)(4-3x)⁶ dx

Question 2 :

(3/x)+[m/(4x+1)]-2(5-2x)⁵

Solution :

Now we are going to integrate the given function

=  ∫(3/x)+[m/(4x+1)]-2(5-2x)⁵ dx

=  3∫(1/x) dx + m∫[1/(4x+1)] dx - 2∫(5-2x)⁵ dx

=  3log x+(m/4) log (4x+1)+(1/6)(5-2x)⁶+C

Question 3 :

4 - 5/(x+2) + 3 cos 2x

Solution :

Now we are going to integrate the given function

= ∫(4 - 5/(x + 2) + 3 cos 2x) dx

=  4∫dx - 5∫[1/(x+2)] dx + 3∫cos 2 x dx

=  4x-5 log (x+2) + (3/2) sin 2x + C

Question 4 :

3e^7x-4sec(4x+3)tan(4x+3)+11/x⁵

Solution :

Now we are going to integrate the given function

=  ∫(3e^7x-4sec(4x+3)tan(4x+3)+11/x⁵) dx

=  3∫e^7x dx - 4∫sec(4x+3)tan(4x+3) dx + 11∫x^-5 dx

=  (3/7) e^7x - sec (4x+3) - (11/4x⁴) + C

Question 5 :

pcosec²(px-q)-6(1-x)⁴+4e^(3-4x)

Solution :

Now we are going to integrate the given function

=  ∫[pcosec²(px-q)-6(1-x)⁴+4e^(3-4x) ] dx

=  p∫cosec² (px-q) dx - 6∫(1-x)⁴ dx + 4 ∫e^(3-4x) dx

=  -cot (px-q)- (6/5)(1-x)⁵ -e^(3-4x) + C

Question 6 :

(2x-5) (36+4x)

Solution :

=  ∫(2x-5) (36+4x) dx

=  ∫[2x(36) + 2x(4x) - 5(36) - 5(4x)] dx

= ∫[52x+8x²-180] dx

= ∫52 x dx + ∫8x² dx - ∫180 dx

= 52 ∫ x dx  + 8 ∫ x² dx - 180 ∫ dx

=  52x²/2 + 8x³/3 - 180x + C

=  26x² + 8x³/3 - 180x + C

Question 7 :

(1 + x³)²

Solution :

(a+b)²  =  a²+2ab+b²

=  ∫(1+x³)² dx

=  ∫(1+2x³+x⁶) dx

=  ∫1 dx + ∫2x³ dx + ∫x⁶ dx

=  x+2∫x³dx + ∫x⁶ dx

=  x+2x⁴/4 + x⁷/7 + C

=  x+x⁴/2+x⁷/7+C

= ( x⁷/7) + (x⁴/2) + x + C

Question 8 :

(x³+4x²-3x+2)/x²

Solution :

=  ∫[(x³+4x²-3x+2)/x²] dx

=  ∫x dx + ∫4 dx - ∫(3/x) dx + ∫(2/x²) dx

=  x²/2 + 4 x - 3 log x - 2/x + C

Question 9 :

(x⁴ - x² + 2)/(x + 1)

Solution :

Now we are going to integrate the given function

=  ∫ (x⁴-x²+2)/(x+1) dx

=  ∫ [(x³-x²) + 2/(x+1)] dx

=  ∫(x³-x²) dx + ∫ 2/(x+1) dx

=  (x⁴/4) - (x³/3) + 2 log (x + 1) + C

Question 10 :

(1+x)²/√x

Solution :

=  ∫[(1+x)²/√x] dx

(a + b)²  =  a²+2ab+b²

=  ∫[(1+2x+x²)/√x] dx

=  ∫[(1/√x) dx + ∫(2x/√x) dx + ∫ x²/√x dx

=  2√x + 2 (2/3) x √x + (2/5) x²√x + C

=  2√x + (4/3) x √x + (2/5) x²√x + C

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