Let (\[t]) denote the greatest integraleger less than or equ... | Mathematics | SolveeIt

Question

Let $[t]$ denote the greatest integer less than or equal to t Then the value of the integral $\int\limits_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$ is equal to

$\frac{52(1- e )}{ e }$

$\frac{52}{ e }$

$\frac{52(2+ e )}{ e }$

$\frac{104}{ e }$

Answer

$\frac{52}{ e }$

Explanation

Solution

−3∫101([sinπx]+e[cos2πx])dx
520∫2([sinπx]+e[cos2πx])dt
π520∫2π([sint]+e[cos2t])dt
π520∫2π([sint]dt+0∫2πe[cos2t]dt)
I1=0∫2π[sint]dt
Using King
I1=0∫2π[−sint]dt
2I1=0∫2π(−1)dt=−2π
I1=−π
I2=20∫πe[cos2t]dt
=2.20∫π/2e[cos2t]dt
=4(0∫π/4e0⋅dt+π/4∫π/2e−1dt)
4(4π+e−1(4π))=π(1+e−1)
I=π52(−π+π+πe−1)=e52

Mathematics Question on Definite Integral

Solution