Integral Calculus: Definite and Indefinite Integrals, Reduction Formulas and Areas Under Curves

The value of $\int \cos^{-1}(2x)\,dx$ is given by :

The area under the curve $y = \cos x$ on $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is given by :

The value of $\iint_R y^2 x \,dydx$ over the rectangle $R = \{(x, y) : -3 \le x \le 2, 0 \le y \le 1\}$ is given by :

The value of $\int_0^6 f(x)\,dx$ if $f(x) = \begin{cases} x^2, & x < 2 \\ 3x-2, & x \ge 2 \end{cases}$ is given by :

If $\Gamma(n) = \int_0^\infty e^{-x}x^{n-1}\,dx, (n > 0)$, then the value of $\int_0^{\pi/2}\sin^n x\,dx$ is given by :

The value of $\int x\tan^{-1}x \, dx$ is given by :

The value of $\int_0^{\frac{\pi}{2}} \sin^2\frac{x}{2} \cos^2\frac{x}{2} \, dx$ is given by :

The value of $\int_0^\pi \int_0^{\cos y} x \sin y \, dx \, dy$ is given by :

If $f$ and $g$ are integrable on $[a, b]$ and $f(x) \ge g(x)$ for all $x$ in $[a, b]$, then :

Suppose that a particle moves on a coordinate line so that its velocity at time $t$ is $v(t) = t^2 - 2t\text{ m/s}$, then the displacement of the particle during the time interval $0 \le t \le 3$ is given by :

Given $\Gamma(n) = \int_0^\infty e^{-x} x^{n-1} dx$, $(n > 0)$, then the value of $\int_0^{\pi/2} \cos^n x \, dx$ is given by :