If a and b are unit vectors and θ is the angle between them, then the value of cosθ2 is
12a + b
12a - b
a - ba + b
a + ba - b
∫xsec2xdx is equal to
xtanx + logsecx + c
x22secx + logcosx + c
xtanx + logcosx + c
tanx + logcosx + c
∫te3t2dt is equal to
16e3t2 + c
- 16e3t2 + c
16e- 3t2 + c
- 16e- 3t2 + c
∫0πlogsin2xdx is equal to
2πloge12
πloge2
π2loge12
None of these
A.
Let I = ∫0πlogsin2xdx = 2∫0πlogsinxdx = 4∫0π2logsinxdx ...i⇒ I = 4∫0π2logsinπ2 - xdx = 4∫0π2logcosxdx ...iiOn adding Eqs. (i) and (ii), we get 2I = 4∫0π2logsinxcosxdx⇒ I = 2∫0π2logsin2x - log2dx = 2∫0π2logsin2xdx - 2xlog20π2 = 22∫0πlogsint - 2 . π2loge2∵ Putting 2x = t
= 2∫0π2logsinxdx - πloge2⇒ I = 2 . I4 - πloge2 ∵ using Eq. (i)⇒ I - I2 = - πloge2⇒ I = - 2πloge2 = 2πloge12
∫dxxxn + 1 is equal to
1nlogxnxn + 1 + c
1nlogxn + 1xn + c
logxnxn + 1 + c
The area bounded by the curves y2 - x = 0 and y - x2 = 0, is
7/3 sq unit
1/3 sq unit
5/3 sq unit
1 sq unit
∫02x2dx is equal to
2 - 2
2 + 2
2 - 1
- 2 - 3 + 5
Area between the curve y = cos(x) and X - axis, when 0 ≤ x ≤ 2π, is
0 sq units
2 sq units
3 sq units
4 sq units
In = ∫0π4tannxdx, then limn→∞nIn + In + 2 equals
1/ 2
4 sq uits
The order of the differential equation
ydydx = xdydx + dydx3 is
1
2
3
4