Integration

The area (in sq. units) of the region

The area (in sq. units) of the region {(x, y) : x ≥ 0, x + y ≤ 3, x² ≤ 4y and y ≤ 1 + √x } is

59/12
3/2
7/3
5/2

The integral is equal to: #03

The integral ₀∫^π √(1 + 4 sin²x/2 - 4 sinx/2) dx equals

4√3 - 4
4√3 - 4 - π/3
2π/3 - 4 - 4√3
π - 4

The integral is equal to: #02

The integral ∫(1 + x - 1/x)e^{x + 1/x} dx is equal to

xe^{x + 1/x} + c
(x + 1)e^{x + 1/x} + c
(x - 1)e^{x + 1/x} + c
-xe^{x + 1/x} + c

If the integral ∫ (5 tan x / tan x − 2)dx = x + a ln |sin x - 2 cos x| + k

If the integral ∫ (5 tan x / tan x − 2)dx = x + a ln |sin x - 2 cos x| + k, then a is equal to

-1
2
-2
1

If g(x) = 0∫x cos4t dt, then g(x + π) equals

If g(x) = ₀∫^x cos4t dt, then g(x + π) equals

g(x) - g(π)
g(x) / g(π)
g(x) + g(π)
g(x).g(π)

0∫π [cot x] dx, where [.] denotes the greatest integer function

₀∫^π [cot x] dx, where [.] denotes the greatest integer function, is equal to

-π/2
1
-1
π/2

∫ |sin x| dx is

₀∫^10π |sin x| dx is

8
10
18
20

The area of the region bounded by the parabola (y – 2)^2 = x – 1, the tangent to the parabola

The area of the region bounded by the parabola (y – 2)² = x – 1, the tangent to the parabola at the point (2, 3) and the x-axis is

3
6
9
12

The integral is equal to: #01

The integral \( \int \dfrac{2x^{12}+5x^9}{(x^5+x^3+1)^3} dx \) is equal to:

\( \dfrac{-x^5}{(x^5+x^3+1)^2} + C \)
\( \dfrac{x^{10}}{2(x^5+x^3+1)^2} + C \)
\( \dfrac{x^5}{2(x^5+x^3+1)^2} + C \)
\( \dfrac{-x^{10}}{2(x^5+x^3+1)^2} + C \)