Question Statement

Evaluate the integral:

$\int [1 - 8 cosec^{2} (2 x)] d x$

Background and Explanation

This problem requires applying the linearity property of integrals to separate the expression into simpler terms, along with the standard integral formula for $cosec^{2} (a x)$ . The solution can be verified by differentiation using the chain rule.

Solution

Let $I$ denote the integral:

$I = \int (1 - 8 cosec^{2} 2 x) d x$

Using the linearity property of integration, we split this into two separate integrals:

$I = \int 1 d x - 8 \int cosec^{2} 2 x d x$

Evaluating each part:

The integral of $1$ with respect to $x$ is $x$
For the second integral, we apply the standard formula $\int cosec^{2} (a x) d x = \frac{- c o t ( a x )}{a}$

With $a = 2$ :

$I = x - 8 (\frac{- c o t 2 x}{2}) + c$

Simplifying the coefficient ( $- 8 \times - \frac{1}{2} = 4$ ):

$I = x + 4 cot 2 x + c$

Verification by Differentiation

To confirm the answer, we differentiate the result:

$\frac{d}{d x} (x + 4 cot 2 x + c)$

Applying the sum rule and differentiating term by term:

$= \frac{d}{d x} (x) + \frac{d}{d x} (4 cot 2 x) + \frac{d}{d x} (c)$

$= 1 + 4 \frac{d}{d x} (cot 2 x) + 0$

Using the chain rule for $cot (2 x)$ , where the derivative of $cot (u)$ is $- cosec^{2} (u)$ and the derivative of $2 x$ is $2$ :

$= 1 + 4 (- cosec^{2} 2 x) \cdot 2$

$= 1 - 8 cosec^{2} 2 x$

This matches the original integrand, confirming the solution is correct.

Key Formulas or Methods Used

Linearity of Integration: $\int [f (x) - g (x)] d x = \int f (x) d x - \int g (x) d x$
Standard Integral: $\int cosec^{2} (a x) d x = - \frac{c o t ( a x )}{a} + C$
Chain Rule: $\frac{d}{d x} [cot (a x)] = - a cosec^{2} (a x)$
Basic Integral: $\int 1 d x = x + C$

Summary of Steps

Apply linearity to split the integral: $\int 1 d x - 8 \int cosec^{2} (2 x) d x$
Integrate the first term to obtain $x$
Use the standard formula for $\int cosec^{2} (a x) d x$ with $a = 2$ to get $- \frac{c o t ( 2 x )}{2}$
Simplify the constants: $- 8 \times (- \frac{c o t ( 2 x )}{2}) = 4 cot (2 x)$
Combine and add constant of integration: $x + 4 cot (2 x) + c$
Verify by differentiation using the chain rule to ensure the derivative equals $1 - 8 cosec^{2} (2 x)$

Question Statement

Evaluate the integral:

$\int [1 - 8 cosec^{2} (2 x)] d x$

Background and Explanation

Solution

Let $I$ denote the integral:

$I = \int (1 - 8 cosec^{2} 2 x) d x$

Using the linearity property of integration, we split this into two separate integrals:

$I = \int 1 d x - 8 \int cosec^{2} 2 x d x$

Evaluating each part:

The integral of $1$ with respect to $x$ is $x$
For the second integral, we apply the standard formula $\int cosec^{2} (a x) d x = \frac{- c o t ( a x )}{a}$

With $a = 2$ :

$I = x - 8 (\frac{- c o t 2 x}{2}) + c$

Simplifying the coefficient ( $- 8 \times - \frac{1}{2} = 4$ ):

$I = x + 4 cot 2 x + c$

Verification by Differentiation

To confirm the answer, we differentiate the result:

$\frac{d}{d x} (x + 4 cot 2 x + c)$

Applying the sum rule and differentiating term by term:

$= \frac{d}{d x} (x) + \frac{d}{d x} (4 cot 2 x) + \frac{d}{d x} (c)$

$= 1 + 4 \frac{d}{d x} (cot 2 x) + 0$

Using the chain rule for $cot (2 x)$ , where the derivative of $cot (u)$ is $- cosec^{2} (u)$ and the derivative of $2 x$ is $2$ :

$= 1 + 4 (- cosec^{2} 2 x) \cdot 2$

$= 1 - 8 cosec^{2} 2 x$

This matches the original integrand, confirming the solution is correct.

Key Formulas or Methods Used

Linearity of Integration: $\int [f (x) - g (x)] d x = \int f (x) d x - \int g (x) d x$
Standard Integral: $\int cosec^{2} (a x) d x = - \frac{c o t ( a x )}{a} + C$
Chain Rule: $\frac{d}{d x} [cot (a x)] = - a cosec^{2} (a x)$
Basic Integral: $\int 1 d x = x + C$

Summary of Steps

Apply linearity to split the integral: $\int 1 d x - 8 \int cosec^{2} (2 x) d x$
Integrate the first term to obtain $x$
Use the standard formula for $\int cosec^{2} (a x) d x$ with $a = 2$ to get $- \frac{c o t ( 2 x )}{2}$
Simplify the constants: $- 8 \times (- \frac{c o t ( 2 x )}{2}) = 4 cot (2 x)$
Combine and add constant of integration: $x + 4 cot (2 x) + c$
Verify by differentiation using the chain rule to ensure the derivative equals $1 - 8 cosec^{2} (2 x)$

3.2 Q-3

Question Statement

Background and Explanation

Solution

Verification by Differentiation

Key Formulas or Methods Used

Summary of Steps

3.2 Q-3

Question Statement

Background and Explanation

Solution

Verification by Differentiation

Key Formulas or Methods Used

Summary of Steps