Let $I = \int_{10}^{19} \frac{\sin x}{1+x^{6}} dx$. Then,

If $\int_{n}^{n+1} f(x) dx = n^2 + n$ for all $n \in I$,then the value of $\int_{-3}^{3} f(x) dx$ is equal to

Let $\alpha \in (0,1)$ and $\beta = \log_{e}(1-\alpha)$. Let $P_n(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \dots + \frac{x^n}{n}$ for $x \in (0,1)$. Then the integral $\int_{0}^{\alpha} \frac{t^{50}}{1-t} dt$ is equal to

The value of $\int_3^5 {\frac{{{x^2}}}{{{x^2} - 4}}\,dx} $ is

Let $f(x) + 2f\left(\frac{1}{x}\right) = x^2 + 5$ and $2g(x) - 3g\left(\frac{1}{x}\right) = x$ for $x > 0$. If $\alpha = \int_1^2 f(x) dx$ and $\beta = \int_1^2 g(x) dx$,then the value of $9\alpha + \beta$ is:

Prove that $\int_{1}^{3} \frac{dx}{x^{2}(x+1)} = \frac{2}{3} + \log \frac{2}{3}$.