If $\int_{0}^{1}(5x^{2}-3x+k)dx=0$,then $k=$

If the real part of the complex number $(1-\cos \theta+2 i \sin \theta)^{-1}$ is $\frac{1}{5}$ for $\theta \in(0, \pi)$,then the value of the integral $\int_{0}^{\theta} \sin x \,dx$ is equal to:

If for all real triplets $(a, b, c)$,$f(x) = a + bx + cx^2$,then $\int_{0}^{1} f(x) dx$ is equal to:

The value of $\int_{0}^{\pi / 2} \frac{\cos 3 x+1}{2 \cos x-1} d x$ is

$\int_{\pi / 4}^{\pi / 3} \frac{\cos x-\sin x}{\sin 2 x} d x=$

Let $f$ be a polynomial function such that $f(x^{2}+1)=x^{4}+5x^{2}+2$ for all $x \in \mathbb{R}.$ Then $\int_{0}^{3} f(x) dx$ is equal to