Answer:

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)ⁿ 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.