Definite Integrals MCQ Quiz - Objective Question with Answer for Definite Integrals - Download Free PDF

Analysis:

Consider 

Put sin θ = t

cos θ dθ = dt

If θ = 0 to  then t = 0 to t = 1

Now, 

Concept:

Trigonometric Ratio Fundamental Identities

Integration by parts

when u and v are functions of x.

Integral of standard function 

The derivative of standard function

Calculation:

Given:

we have 

where u = x and 

Integration by parts

when u and v are functions of x.

Integration by parts

when u and v are functions of x.

∫x × sec² x dx = x tan x - ∫ tan x dx

∫x × sec2 x dx = x tan x - (-log (cos x))

∫ x × sec2 x dx = x tan x + log (cos x)

Concept:

Arc length or curve length is the distance between two points along the section of the curve. Determining the length of an irregular section of the arc is termed as rectification of the curve.

The length of the curve y = f(x) from x = a to x = b is given as:

or,

If the curve is parametrized in the form x = f(t) and y = g(t) with the parameter t going from a to b then

Calculation:

Now, the arc length(l) is

Explanation:

The given integral is an improper integral of 1^st kind.

I = log [log (∞)] – log [log (2)]

∴ I = ∞

Given integral is divergent and diverges to ∞

Additional Information

An improper integral of first kind is when integral limits have -∞ or +∞ or both.

An improper integral of second kind is when integral limits are finite but function is infinite at some value between those limits.

Concept:

Using Integration by parts using ILATE

Calculation:

Concept:

We know that,

By Parts method

Where, u, v should follow the ILATE sequence.[I= Inverse, L= Logarithmic, A= Algebraic, T= Trigonometric, E= Exponential terms]

Calculation:

Given:

From the given Equation, 

u = ln(x), v = √x

Now,

∴         

Explanation

Given,

 Function f(x) = x e^|x|

 Integral is -1 to 1.

If f(-x) = f(x) then the function is said to be even function

 If f(-x) = - f(x) then the function is said to be odd function.

 f(-x) = -x e^|-x| = -x e|x| = - f(x)

 ∴ The given function is an odd function.

 For an odd function:

  = 0

For a even function

  = 2 × 

 Now, as the function is odd

  = 0

Explanation:

I = [-e^-0 + e¹] + [e¹ - e⁰]

I = -1 + e¹ + e - 1

Concept:

So it represents an equation of circle in x-y plane.

Given 0 ≤ u ≤ 1

So, 0 ≤ x ≤ 1,  0 ≤ y ≤ 1

i.e., 0 ≤ θ ≤ π/2

So we get a quarter circle in x-y plane and by revolving it 360°, we get a hemisphere.

Area of hemi-sphere = 2π(1)² = 2π

Concept:

Volume of the solid of rotation obtained by rotating the shaded area by 360° around the x-axis is asked,

there is a direct relation for this;

 …1)

Calculation:

Given area;

Using (1); Volume of solid of rotation can be calculated by:

Key points:

In the given problem, the volume is generated by revolving the area by 360° about the x-axis.

But if rotation/revolution is about the y-axis, then the volume of solid of rotation is calculated by:

 …2)

So, always be careful about which axis rotation is asked.

Depending upon that, you should use either 1) or 2).

f(x) = x – [x]

for 0.25

for 1

Given x as non- zero,

       ---(1)

Consider x as 1/x

       ---(2)

Multiply equation 1 by a and 2 by b and subtract both

Concept:

Calculation:

⇒ Let, I =     ----- equation(1)

⇒ I = 

⇒ I =      ---- equation(2)

On adding equation (1) and (2)

⇒ 2I = 

⇒ 2I = 

⇒ 2I = 

⇒ I = 

∴ The value of  is 

Concept:

Following steps to solve the equation

1. To remove the modulus
2. To keep sin x positive in the interval 0 to π to 2π and to keep the sin x negative in the interval.This is because x in the above equation is always positive but the value sinx changes in the two mentioned intervals.

Explanation:

Solving the equation as per the steps,

Keeping u = x, du = dx, dv = sinxdx, so v = - cosx,

Now, repeating the same with , we get -3

Hence, π - (-3π) = 4π

Therefore, k = 4

Concept:

Use integration by parts for solving this problem.

Calculation: