SATHEE: Indefinite Integration 2 Question 4

Indefinite Integration 2 Question 4

4.. Show that $\int_{0}^{n π + v} | \sin x | d x = 2 n + 1 - \cos v$ , where $n$ is a positive integer and $0 \leq v < π$ .

(1994, 4M)

Show Answer

Answer:

Correct Answer: 4. (4)

Solution:

$\int_{0}^{n π + v} | \sin x | d x = \int_{0}^{π} | \sin x | d x + \int_{π}^{2 π} | \sin x | d x + \dots$

$+ \int_{(n - 1) π}^{n π} | \sin x | d x + \int_{n π}^{n π + v} | \sin x | d x$

$= \sum_{r = 1}^{n} \int_{(r - 1) π}^{r π} | \sin x | d x + \int_{n π}^{n π + v} | \sin x | d x$

Now to solve, $\int_{(r - 1) π}^{r π} | \sin x | d x$ , we have

$\begin{aligned} x & = (r - 1) π + t \\ \Rightarrow \sin x & = \sin [(r - 1) π + t] = (- 1)^{r - 1} \sin t \end{aligned}$

and when $x = (r - 1) π, t = 0$ and when

$x = r π, t = π$

$∴ \int_{(r - 1) π}^{r π} | \sin x | d x = \int_{0}^{π} | (- 1)^{r - 1} \sin t | d t$

$\begin{aligned} = \int_{0}^{π} | \sin t | d t = \int_{0}^{π} \sin t d t \\ = [- \cos t]_{0}^{π} = - \cos π + \cos 0 = 2 \end{aligned}$

Again, $\int_{n π}^{n π + v} | \sin x | d x$ , putting $x = n π + t$

Then, $\int_{n π}^{n π + v} | \sin x | d x = \int_{0}^{v} | (- 1)^{n} \sin t | d t = \int_{0}^{v} \sin t d t$

$= [- \cos t]_{0}^{v} = - \cos v + \cos 0 = 1 - \cos v$

$∴ \int_{0}^{n π + v} | \sin x | d x = \sum_{r = 1}^{n} \int_{(r - 1) π}^{r π} \ln \sin x | d x + \int_{n π}^{n π + v} | \sin x ∣ d x$

$\begin{aligned} = \sum_{r = 1}^{n} 2 + \int_{n π}^{n π + v} | \sin x | d x \\ = 2 n + 1 - \cos v \end{aligned}$

Indefinite Integration 2 Question 4

4.. Show that $\int_{0}^{n π + v} | \sin x | d x = 2 n + 1 - \cos v$ , where $n$ is a positive integer and $0 \leq v < π$ .

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Indefinite Integration 2 Question 4

4.. Show that ∫0nπ+v|sin⁡x|dx=2n+1−cos⁡v, where n is a positive integer and 0≤v<π.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

4.. Show that $\int_{0}^{n π + v} | \sin x | d x = 2 n + 1 - \cos v$ , where $n$ is a positive integer and $0 \leq v < π$ .