Definite Integration Maths - Definite Integration

$\int_{-2}^3|x-2| d x$

$\int_0^4\left[\sqrt{x^2+2 x+3}\right]^{-1} d x$

$\int_0^{\pi / 2} x \cdot \sin x \cdot d x$

$\int_0^1 \frac{1}{\sqrt{3+2 x-x^2}} \cdot d x$

$\int_0^1 \frac{x^2-2}{x^2+1} \cdot d x$

$\int_0^\pi\left(\sin ^{-1} x+\cos ^{-1} x\right)^3 \sin ^3 x d x$

$\int_0^{\pi / 2}[2 \log (\sin x)-\log (\sin 2 x)] d x$

$\int_0^1\left(\frac{1}{1+x^2}\right) \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x$

$\int_0^{\pi / 4} \frac{\tan ^3 x}{1+\cos 2 x} d x$

If $f ( x )= a + bx +c x^2$, show that $\int_0^1 f(x) d x=\frac{1}{6}\left[f(0)+4 f\left(\frac{1}{2}\right)+f(1)\right]$

If $\int_0^k \frac{1}{2+8 x^2} \cdot d x=\frac{\pi}{16}$, find $k$.

If $\int_a^a \sqrt{x} d x=2 a \int_0^{\pi / 2} \sin ^3 x d x$, find the value of $\int_a^{a+1} x d x$.

$\int_0^1 \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x$

$\int_{\pi / 5}^{3 \pi / 10} \frac{\sin x}{\sin x+\cos x} d x$

$\int_0^{\pi / 4} \frac{\cos 2 x}{1+\cos 2 x+\sin 2 x} d x$

$\int_0^\pi \frac{x}{1+\sin ^2 x} d x$

$\int_0^\pi x \cdot \sin x \cdot \cos ^4 x d x$

$\int_{\pi / 4}^{\pi / 2} \frac{\cos \theta}{\left[\cos \frac{\theta}{2}+\sin \frac{\theta}{2}\right]^3} d \theta$

$\int_0^1\left(\cos ^{-1} x\right)^2 d x$

$\int_0^{\pi / 2} \frac{\cos x}{3 \cos x+\sin x} d x$