Simple Problems on Integration





The page simple problems on integration we are going to see how to integrate using basic formulas. We have lots of methods in integration. Let us see them one by one. By practicing simple-problems-on-integration we can easily solve difficult problems. 

Example 1:

Integrate x⁶ dx

Solution:

Formula :

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

∫x⁶ dx = x⁽⁶ ⁺ ¹⁾ /(6 + 1) + c

         = x⁷/7 + c



Example 2:

Integrate x ⁻² dx

Solution:

Formula :

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

∫x⁶ dx = x⁽⁻² ⁺ ¹⁾ /(-2 + 1) + c

         = x⁻¹/(-1) + c

         = (-1/x) + c


Example 3:

Integrate √x⁵  dx

Solution:

Formula :

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

∫√x⁵  dx = ∫ (x⁵)^1/2  dx

            = ∫ (x^5/2)  dx

            =  x^[(5/2) + 1)]/[(5/2) + 1)] + C

            =  x^[(5 + 2)/2)]/[(5 + 2)/2)] + C

            =  x^(7/2)/(7/2) + C

            =  (2/7)x^(7/2) + C  simple problems on integration


Example 4:

Integrate Sin x/Cos ² x dx

Solution:

∫Sin x/Cos ² x dx = ∫(Sin x/Cos x) x (1/ Cos x) dx

                        = ∫ tan x  sec x dx

Formula :

∫ sec x tan x dx = sec x  + c

                        = Sec x + c


Example 5:

Integrate Cot x/Sin x dx

Solution:

∫ (Cot x/Sin x) dx = ∫(cot x) x (1/ sin x) dx

                        = ∫ Cot x Cosec x dx

Formula :

∫ Cosec x cot x dx = - Cosec x  + c

                        = - Cosec x + c


Example 6:

Integrate 1/Sin² x dx

Solution:

∫ 1/Sin ²x dx = ∫ Cosec ² x dx

Formula :

∫ Cosec ² x dx = - Cot x  + c

                  =  - Cot x + c


Example 7:

Integrate (3-4x)⁶ dx

Solution:

Formula :

∫ (ax + b) dx = (1/a) (ax + b)⁽ⁿ ⁺ ¹⁾/(n + 1)   + c

∫ (3-4x)⁶ dx = (3-4x)⁽⁶ ⁺ ¹⁾/(6 + 1) (-1/4) + C

                 = (-1/4) (3-4x)⁷/7  + C

                 = (-7/4) (3-4x)⁷ + C


Example 8:

Integrate 1/(3+5x) dx

Solution:

Formula :

1/(ax + b) dx = (1/a) log (ax + b) + c

∫ 1/(3+5x) dx = (1/5) log (3+5x) + C


Example 9:

Integrate Cosec (4x + 3) cot (4x + 3) dx

Solution:

Formula :

∫ Cosec (ax+b)cot (ax+b)dx=-(1/a)Cosec (ax+b) + c

∫ Cosec (4x + 3) cot (4x + 3) dx = - Cosec (4x + 3) (1/4) + C

                                            = - (1/4) Cosec (4x + 3)  + C 


Example 10:

Integrate Sec (ax + b) tan (ax + b) dx

Solution:

Formula :

∫ sec (ax + b) tan (ax + b) dx = sec (ax + b) + c

∫ Sec (ax + b) tan (ax + b) dx = Sec (ax + b) (1/a) + C

                                          = (1/a) Sec (ax + b) (1/a) + C

These are the example problems in the topic integration. simple problems on integration

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