Maths Solve the Following Question.(3 Marks)

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 $\int_a^{a+1} x d x$.

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

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

$\int_0^{\pi / 2} \frac{1}{6-\cos x} d x$

$\int_1^{\infty} \frac{1}{\sqrt{x}(1+x)} d x$

$\int_0^1 t^5 \sqrt{1-t^2} d t$

$\int_0^1 \frac{1}{1+\sqrt{x}} d x$

$\int_0^1 x \cdot \tan ^{-1} x \cdot d x$

$\int_0^{\pi / 4} \sin ^4 x \cdot d x$

$\int_3^5 \frac{1}{\sqrt{2 x+3} \sqrt{2 x-3}} \cdot d x$

$\int_{-\pi / 4}^{\pi / 4} \frac{1}{1-\sin x} \cdot d x$

$\int_{\frac{1}{\sqrt{2}}}^1 \frac{\left(e^{\cos ^{-1} x}\right)\left(\sin ^{-1} x\right)}{\sqrt{1-x^2}} \cdot d x$

$\int_0^{\pi / 2} \sin 2 x \cdot \tan ^{-1}(\sin x) \cdot d x$

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

$\int_0^\pi \frac{1}{3+2 \sin x+\cos x} \cdot d x$

$\int_0^{\pi / 2} \frac{\cos X}{(1+\sin x)(2+\sin x)} \cdot d x$

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

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

$\int_0^{\frac{1}{\sqrt{2}}} \frac{\sin ^{-1} x}{\left(1-x^2\right)^{\frac{3}{2}}} \cdot d x$

$\int_0^3 x^2(3-x)^{\frac{5}{2}} \cdot d x$