MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Integration by Parts

The value of ...

Question

The value of $(\int_{0}^{100 π + v} | sin x | d x) + cos V$ is $(0 \leq V \leq π)$ .

Open in App

Solution

$\int_{0}^{100 π + v} | sin x | d x = \int_{0}^{π} | sin x | d x + \int_{π}^{2 π} | sin x | d x + . . . + \int_{99 π}^{100 π} | sin x | d x + \int_{100 π}^{100 π + v} | sin x | d x$
$= 100 \sum r = 1 \int_{(r - 1) π}^{r π} | sin x | d x + \int_{100 π}^{100 π + v} | sin x | d x$
Now for $\int_{(r - 1) π}^{r π} | sin x | d x$
Let $x = (r - 1) π + t$
$\Rightarrow sin x = sin [(r - 1) π + t] = {(- 1)}^{r - 1} sin t$
When $x = (r - 1) π, t = 0$ and when $x = r π, t = π$
$∴ \int_{(r - 1) π}^{r π} | sin x | d x = \int_{0}^{π} ∣ ∣ {(- 1)}^{r - 1} l i m t ∣ ∣ d t$
$= \int_{0}^{π} | sin x | d t = \int_{0}^{π} sin t d t$
$= {[- cos t]}_{0}^{π} = - cos π + cos 0 = 2$
Again for $\int_{100 π}^{100 π + v} | sin x | d x$
Put $x = 100 π + t$
$\int_{100 π}^{100 π + v} | sin x | d x = \int_{0}^{u} {(- 1)}^{n} sin t d t = \int_{0}^{v} sin t d t$
$∴ \int_{0}^{100 π + v} | sin x | d x = 100 \sum r = 1 \int_{(r, 1) π}^{r π} | sin x | d x + \int_{100 π}^{100 π + v} | sin x | d x$
$= 100 \sum r = 1 2 + \int_{100 π}^{100 π + v} | sin x | d x = 201 - cos v$

Suggest Corrections

thumbs-up

0

Similar questions

\int_{0}^{n π + v} | sin x | d x = 2 n + 1 - cos v

n

is a positive interger and

0 \leq v < π

Q. The value of

n π - \frac{π}{4} \int - \frac{π}{4} | sin x + cos x | d x

Q. The value of

\int_{- \frac{π}{4}}^{n π - \frac{π}{4}} | sin x + cos x | d x

Q. The value of

I = \int_{\frac{- π}{2}}^{\frac{π}{2}} | s i n x | d x

I = n π + λ \int 0 | sin x | d x

n \in N, 0 \leq λ < π

. Then the value of

I

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Integration by Parts

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Integration by Parts

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon