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Important Questions: Integrals | Mathematics (Maths) Class 12 - JEE PDF Download

Q1: Integrate: ∫ sin3 x cos2x dx
Ans: 
Given that, ∫ sin3 x cos2x dx
This can be written as:
∫ sin3 x cos2x dx = ∫ sin2 x cos2x (sin x) dx
=∫(1 – cos2x ) cos2x (sin x) dx —(1)
Now, substitute t = cos x,
Then dt = -sin x dx
Now, equation can be written as:
Thus, ∫ sin3 x cos2x dx = – ∫ (1-t2)t2 dt
Now, multiply t2 inside the bracket, we get
= – ∫ (t2-t4) dt
Now, integrate the above function:
= – [(t3/3) – (t5/5)] + C —(2)
Where C is a constant
Now, substitute t = cos x in (2)
= -(⅓)cos3x +(1/5)cos5x + C
Hence, ∫ sin3 x cos2x dx = -(⅓)cos3x +(1/5)cos5x + C

Q2: Determine the antiderivative F of “f”, which is defined by f (x) = 4x3 – 6, where F (0) = 3
Ans: 
Given function: f (x) = 4x3 – 6
Now, integrate the function:
∫4x3 – 6 dx = 4(x4/4)-6x + C
∫4x3 – 6 dx = x4 – 6x + C
Thus, the antiderivative of the function, F is x4 – 6x + C, where C is a constant
Also, given that, F(0) = 3,
Now, substitute x = 0 in the obtained antiderivative function, we get:
(0)4 – 6(0) + C = 3
Therefore, C = 3.
Now, substitute C = 3 in antiderivative function
Hence, the required antiderivative function is x4 – 6x + 3.

Q3: Determine ∫tan8x sec4 x dx
Ans:
Given: ∫tan8x sec4 x dx
Let I = ∫tan8x sec4 x dx — (1)
Now, split sec4x = (sec2x) (sec2x)
Now, substitute in (1)
I = ∫tan8x (sec2x) (sec2x) dx
= ∫tan8x (tan2 x +1) (sec2x) dx
It can be written as:
= ∫tan10x sec2 x dx + ∫tan8x sec2 x dx
Now, integrate the terms with respect to x, we get:
I =( tan11 x /11) + ( tan9 x /9) + C
Hence, ∫tan8x sec4 x dx = ( tan11 x /11) + ( tan9 x /9) + C

Q4: Integrate the given function using integration by substitution: 2x sin(x

 + 1) with respect to x.
Ans: 
Given function: 2x sin(x2+ 1)
We know that, the derivative of x2 + 1 is 2x.
Now, use the substitution method, we get
x2 + 1 = t, so that 2x dx = dt.
Hence, we get ∫ 2x sin ( x2 +1) dx = ∫ sint dt
= – cos t + C
= – cos (x2 + 1) + C
Where C is an arbitrary constant
Therefore, the antiderivative of 2x sin(x+ 1) using integration by substitution method is = – cos (x2 + 1) + C

Q5: Write the anti-derivative of the following function: 3x2 + 4x3
Ans:
Given: 3x2 + 4x3
The antiderivative of the given function is written as:
∫3x2 + 4x3 dx =  3(x3/3) + 4(x4/4
= x+ x4
Thus, the antiderivative of 3x2+ 4x3 = x3 + x4.

Q6: Evaluate: ∫ 3ax/(b2 + c2x2) dx
Ans: 
To evaluate the integral, I = ∫ 3ax/(b2 + c2x2) dx
Let us take v = b2 + c2x2, then
dv = 2c2x dx
Thus, ∫ 3ax/(b2 +c2x2) dx
= (3ax/2c2x)∫dv/v
Now, cancel x on both numerator and denominator, we get
= (3a/2c2)∫dv/v
= (3a/2c2) log |b+c2x2| + C
Where C is an arbitrary constant.

The document Important Questions: Integrals | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Important Questions: Integrals - Mathematics (Maths) Class 12 - JEE

1. What is the concept of integration in calculus?
Ans. Integration is a fundamental concept in calculus that involves finding the area under a curve. It is the reverse process of differentiation and allows us to find the antiderivative of a function. By integrating a function over a specific interval, we can determine the accumulated change or total value of the function within that interval.
2. How is the definite integral different from the indefinite integral?
Ans. The definite integral and the indefinite integral are two different types of integrals in calculus. The definite integral calculates the area under a curve between two specific points, denoted by definite limits of integration. On the other hand, the indefinite integral, also known as the antiderivative, gives a family of functions that have the same derivative. It is represented by an indefinite integral symbol and requires the addition of a constant of integration.
3. What are the different methods to evaluate integrals?
Ans. There are several methods to evaluate integrals, such as: 1. Integration by substitution: This technique involves substituting a variable or expression to simplify the integral. 2. Integration by parts: This method is based on the product rule of differentiation and involves splitting the integral into two parts. 3. Trigonometric substitution: It is used when the integrand contains square roots of quadratic expressions or other trigonometric functions. 4. Partial fraction decomposition: This approach is used to split a rational function into simpler fractions before integrating.
4. How can the definite integral be interpreted geometrically?
Ans. Geometrically, the definite integral represents the area under a curve between two points on the x-axis. It can also be interpreted as the accumulation of infinitesimally small areas between the curve and the x-axis over a given interval. The value of the definite integral provides the net area, taking into account the areas above and below the x-axis. If the integral value is positive, it represents the area above the x-axis, and if it is negative, it represents the area below the x-axis.
5. What are the practical applications of integration?
Ans. Integration has numerous practical applications in various fields, including physics, engineering, economics, and computer science. Some examples include: 1. Finding the area and volume of irregular shapes or objects. 2. Calculating the displacement, velocity, and acceleration of objects in physics. 3. Determining the total revenue or cost of a business over a specific period in economics. 4. Analyzing data and solving differential equations in computer science. 5. Predicting population growth or decay using mathematical models.
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