## Properties of Integrals

In this page Properties of Integrals we are going to see two properties.

Property 1:

if k is constant then  ∫ k f(x) dx = K ∫ f(x) dx

Property 2:

if "f(x) and "g(x)" are any two functions then

∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

Now let us see some example problems based on the above properties.

Example 1:

Integrate 5 x⁶ + 2/x³ dx with respect to x

Solution:

= ∫ (5 x⁶ + 2/x³ ) dx

= ∫ (5 x⁶ ) dx + ∫ ( 2/x³ ) dx

= 5 ∫ x⁶  dx + 2 ∫ 1/x³  dx

= 5 ∫ x⁶  dx + 2 ∫ x⁻³  dx

= 5  x⁽⁶ ⁺ ¹⁾/(6 + 1)  + 2  x⁽⁻³ ⁺ ¹⁾/(-3 +1)

= 5  (x⁷/7)  + 2  (x⁻ ²/-2) + C

= (5/7)  x⁷  -  x⁻ ² + C

= (5/7)  x⁷  -  1/x² + C

= 5x⁷/7  -  1/x² + C

Example 2:

Integrate 4/(3 + 5x) + 3 cos 2x + 8 with respect to x

Solution:

= ∫ [ 4/(3 + 5x) + 3 cos 2x + 8 ] dx

= ∫ [4/(3 + 5x)] dx + ∫ (3 cos 2x) dx + ∫ 8 dx

= 4 ∫ [1/(3 + 5x)] dx + 3 ∫ (cos 2x) dx + 8 ∫ dx

= 4 (1/5) log (3 + 5x) + 3 (- Sin 2x)/2 + 8 x + C

= (4 /5) log (3 + 5x) - (3 Sin 2x)/2 + 8 x + C

= (4 /5) log (3 + 5x) - (3/2) Sin 2x + 8 x + C

Example 3:

Integrate 2x⁴ + 3 (2x + 3)⁴ -  sin 2x + 5/x³ with respect to x

Solution:

∫ (2x⁴ + 3 (2x + 3)⁴ -  sin 2x + 5/x³) dx

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

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

=  2[x⁽⁴ ⁺ ¹⁾/(4+1)]+3[(2x+3)⁽⁴ ⁺ ¹⁾/(4+1)](1/2)-(-cos 2x)(1/2)+5 x⁽⁻³ ⁺ ¹⁾/(-3+1)+C

=  2[x⁵/5]+3[(2x+3)⁵/5](1/2)+ (1/2) cos 2x+5 x⁻²/-2 + C

=  2x⁵/5+(3/5) (1/2) (2x+3)⁵+ (cos 2x/2) - 5/(2 x²) + C

=  2x⁵/5+(3/10)  (2x+3)⁵+ (cos 2x/2) - 5/(2 x²) + C

Related pages

Properties of Integrals to Integration