$\int_1^e \frac{1 + \log x}{x} \, dx = $

The value of the definite integral,$\int\limits_0^{\sqrt {\ln \left( {\frac{\pi }{2}} \right)} } {\cos \left( {{e^{{x^2}}}} \right)} \cdot 2x {e^{{x^2}}}dx$ is

Assume that $f$ is continuous everywhere,then $\frac{1}{c}\int_{ac}^{bc} {f\left( {\frac{x}{c}} \right)} \,dx = $

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

Evaluate the definite integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x+\cos x}{\sqrt{\sin 2 x}} d x$.

$\int_0^4 \frac{1}{1+\sqrt{x}} \, dx = \dots$