List of top Mathematics Questions on Definite Integral asked in AP EAMCET

The value of
\[ \int_0^1 a^k x^k\,dx \] is

Let \(\alpha\) and \(\beta\), \((\alpha\lt \beta)\), are roots of \(18x^2-9\pi x+\pi^2=0\), \(f(x)=x^2\), \(g(x)=\cos x\). Then \(\displaystyle \int_{\alpha}^{\beta} x(gof(x))dx=\)

\[ \int_{0}^{\pi} x\left(\sin^2(\sin x)+\cos^2(\cos x)\right)dx= \]

\[ \lim_{n\to\infty} \left( \frac{1}{1+n^5} + \frac{2^4}{2^5+n^5} + \frac{3^4}{3^5+n^5} +\cdots+ \frac{n^4}{n^5+n^5} \right) = \]