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

Related pages





Featured Categories 

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test