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NDA Exam Mathematics Study Material

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Integration and Differential Equations

If f(x) and g(x) are two real value functions such that∫ f(x)dx and∫ g(x)dx exist, then

a) ∫k f(x)dx = k∫ f(x)dx
b) ∫k f(x)dx - k∫ f(x)dx
c) ∫k f(x)dx + k∫ f(x)dx
d) All of these

✔Answer

Correct Answer Is : ∫k f(x)dx = k∫ f(x)dx

Solution Is :

If∫ f(x)dx = F(x), then∫ f (ax + b)dx =

a) 1/a F(ax + b) + c; a?0
b) 1/a F(ax + b) + c; a=0
c) A F(ax + b) + c; a?0
d) F(ax + b) + c; a?0

✔Answer

Correct Answer Is : 1/a F(ax + b) + c; a?0

Solution Is :

Evaluate: ∫(3x + 4)^3 dx

a) (3x + 4)^4
b) (3x + 4)^2
c) (3x + 4)^4/12
d) (3x + 4)^4/4

✔Answer

Correct Answer Is : (3x + 4)^4/12

Solution Is :

Evaluate: ∫cos^3 x.dx

a) 3sin x /4 + c
b) 3sin x /4 + 3sin 3x /4 + c
c) 3sin x /4 + 3sin 3x /12
d) 3sin x /4 + 3sin 3x /12 + c

✔Answer

Correct Answer Is : 3sin x /4 + 3sin 3x /12 + c

Solution Is :

True or false: if a and b are any non-zero real number,∫(e^(ax + b))dx=e^(ax + b)

a) TRUE
b) FALSE
c) Maybe
d) None of these

✔Answer

Correct Answer Is : FALSE

Solution Is :

Evaluate: ∫sin 7x.cos 3x.dx

a) (-cos 10x/20) - (cos4x/8) + c
b) (-cos 10x/20) + (cos4x/8) + c
c) (-cos 10x/20) - (cos4x/4) + c
d) (-cos 10x/2) - (cos4x/8) + c

✔Answer

Correct Answer Is : (-cos 10x/20) - (cos4x/8) + c

Solution Is :

Integrate the function f(x) = (1/ 1 + cos x) with respect to x.

a) Tan x + c
b) Tan x/2 + c
c) Cot x/2 + c
d) Cot x + c

✔Answer

Correct Answer Is : Tan x/2 + c

Solution Is :

Integrate the function f(x) = (tan x / sec x + tan x) with respect to x.

a) Sec x - tan x + x + c
b) Sec x + tan x + x + c
c) Sec x - tan x - x + c
d) Sec x - tan x + c

✔Answer

Correct Answer Is : Sec x - tan x + x + c

Solution Is :

What are the different methods of integration?

a) Integration by substitution
b) Some special integrals
c) Integrals of type ∫(1/ax^2 + bx + c)dx
d) All of these

✔Answer

Correct Answer Is : All of these

Solution Is :

In which method, we reduced the given function to standard form by changing variable x to t, using some suitable substitution x = ∅(t).

a) Integration by substitution
b) Some special integrals
c) Integrals of type ∫(1/ax^2 + bx + c)dx
d) All of these

✔Answer

Correct Answer Is : Integration by substitution

Solution Is :

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NDA Exam Mathematics Study Material

Integration and Differential Equations

If f(x) and g(x) are two real value functions such that∫ f(x)dx and∫ g(x)dx exist, then

Correct Answer Is : ∫k f(x)dx = k∫ f(x)dx

Solution Is :

If∫ f(x)dx = F(x), then∫ f (ax + b)dx =

Correct Answer Is : 1/a F(ax + b) + c; a?0

Solution Is :

Evaluate: ∫(3x + 4)^3 dx

Correct Answer Is : (3x + 4)^4/12

Solution Is :

Evaluate: ∫cos^3 x.dx

Correct Answer Is : 3sin x /4 + 3sin 3x /12 + c

Solution Is :

True or false: if a and b are any non-zero real number,∫(e^(ax + b))dx=e^(ax + b)

Correct Answer Is : FALSE

Solution Is :

Evaluate: ∫sin 7x.cos 3x.dx

Correct Answer Is : (-cos 10x/20) - (cos4x/8) + c

Solution Is :

Integrate the function f(x) = (1/ 1 + cos x) with respect to x.

Correct Answer Is : Tan x/2 + c

Solution Is :

Integrate the function f(x) = (tan x / sec x + tan x) with respect to x.

Correct Answer Is : Sec x - tan x + x + c

Solution Is :

What are the different methods of integration?

Correct Answer Is : All of these

Solution Is :

In which method, we reduced the given function to standard form by changing variable x to t, using some suitable substitution x = ∅(t).

Correct Answer Is : Integration by substitution

Solution Is :

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