Integral Calculus

The value of $\displaystyle\int_0^{\pi} \sin x \, dx$ is

Let $(a, b)$ be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second quadrant. Then the integral $I = \int_a^b \frac{9x^2}{1 + 5^x} \, dx$ is equal to:

$4\int_0^1 \left(\frac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}}\right) dx - 3 \log_e(\sqrt{3})$ is equal to:

The integral $\displaystyle\int_{0}^{\pi}\frac{8x}{4\cos^2x + \sin^2x}\,dx$ is equal to:

The area of the region $\{(x,y):\lvert x-y\rvert \le y \le 4\sqrt{x}\}$ is:

Let f(x)=\int x³\sqrt{3 - x²}\,dx. If 5f(\sqrt{2}) = -4, then f(1) is equal to:

A line passing through the point $A(-2,0)$ touches the parabola $P:\;y^2 = x - 2$ at the point $B$ in the first quadrant. The area of the region bounded by the line $AB$, the parabola $P$, and the $x$-axis is:

If the area of the region $\{(x,y): 1 + x^2 \le y \le \min\{x + 7,\;11 - 3x\}\}$ is $A$, then $3A$ is equal to:

If the area of the region bounded by the curves $y = 4 - \frac{x^2}{4}$ and $y = \frac{x - 4}{2}$ is equal to $\alpha$, then $6\alpha$ equals:

The integral $\int_{-1}^{\frac{3}{2}} \left| \pi^2 x \sin(\pi x) \right| \, dx$ is equal to: