New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Definite Integration question

2002 · Shift 0 · Q80

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Mathematics
Definite Integration
2002 · Shift 0 · Q80

Definite Integration question

2002 · Shift 0 · Q80

JEE MainMathematicsDefinite IntegrationMCQ+4 / −1

\int\limits_0^{10\pi } {\left| {\sin x} \right|dx}

is

A
$20$
B
$8$
C
$10$
D
$18$

View written solutionFree

Correct answer: A

We need to evaluate $I=\int_0^{10\pi}|\sin x|\,dx.$
Recall the periodic behavior of $|\sin x|$ :

$\sin x$ has period $2\pi$ .
$|\sin x|$ has period $\pi$ .

So it is convenient to use the period $\pi$ .

First compute one-period integral: $\int_0^{\pi}|\sin x|\,dx.$ Since $\sin x\ge 0$ on $[0,\pi]$ , we have $\int_0^{\pi}|\sin x|\,dx=\int_0^{\pi}\sin x\,dx.$ Now, $\int_0^{\pi}\sin x\,dx=[-\cos x]_0^{\pi}=(-\cos\pi)-(-\cos0)=1-(-1?)$ Let us compute carefully: $[-\cos x]_0^{\pi}=(-\cos\pi)-(-\cos0)= -(-1)-(-1)=1+1=2.$ Hence, $\int_0^{\pi}|\sin x|\,dx=2.$
The interval $[0,10\pi]$ contains $\frac{10\pi}{\pi}=10$ periods of $|\sin x|$ .

Therefore, $I=10\times 2=20.$

So the correct option is $\boxed{A: 20}.$

More from Definite Integration

$0 \int 2 [x^{2}] d x$ is2002 · MCQ
$\int_{- π}^{π} \frac{2 x ( 1 + s i n x )}{1 + c o s ^{2} x} d x$ is2002 · MCQ
Let [.] denote the greatest integer function. If $0 \int e^{3} [\frac{1}{e ^{x - 1}}] d x = α - lo g_{e} 2$ , then $α^{3}$ is equal to .2025 · Numerical
Let $f : [1, \infty) \to [2, \infty)$ be a differentiable function. If $10 \int_{1}^{1} f (t) dt = 5 x f (x) - x^{5} - 9$ for all $x ⩾ 1$ , then the value of $f (3)$ is :2025 · MCQ
Let $(a, b)$ be the point of intersection of the curve $x^{2} = 2 y$ and the straight line $y - 2 x - 6 = 0$ in the second quadrant. Then the integral $I = \int_{a}^{b} \frac{9 x ^{2}}{1 + 5 ^{x}} d x$ is equal to :2025 · MCQ
$4 \int_{0}^{1} (\frac{1}{3 + x ^{2} + 1 + x ^{2}}) d x - 3 lo g_{e} (3)$ is equal to :2025 · MCQ
Let the domain of the function $f (x) = lo g_{2} lo g_{4} lo g_{6} (3 + 4 x - x^{2})$ be $(a, b)$ . If $\int_{0}^{b - a} [x^{2}] d x = p - q - r, p, q, r \in N, gcd (p, q, r) = 1$ , where $[\cdot]$ is the…2025 · MCQ
The integral $\int_{0}^{π} \frac{8 x d x}{4 c o s ^{2} x + s i n ^{2} x}$ is equal to2025 · MCQ