86391 If \(f(x)\) is continuous at \(x=0\) and \(f(0)=2\), then what is the value of \(\lim _{x \rightarrow 0}\left[\frac{\int_{0}^{x} f(u) d u}{x}\right]\) ?

86392 Let \(f: \mathrm{R} \rightarrow \mathrm{R}\) be a differentiable function such that \(f(3)=3, f^{\prime}(3)=\frac{1}{2}\). Then the value of
\(\lim _{x \rightarrow 3} \int_{3}^{f(x)}\left[\frac{2 t^{3}}{x-3}\right] \text { is }\)

86393 If \(f: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) are continuous functions, then the value of the integral \(\int_{-\pi / 2}^{\pi / 2}(f(x)+f(-x))(g(x)-g(-x)) d x\) is

86394 If \(\int_{0}^{\pi n} f\left(\cos ^{2} x\right) \mathrm{d} x=k \int_{0}^{\pi} f\left(\cos ^{2} x\right) \mathrm{d} x\), then the value of \(k\) is

86395 If \([t]\) denotes the greatest integer \(\leq t\), then the value of \(\frac{3(e-1)}{e} \int_{1}^{2} x^{2} e^{[x]+\left[x^{3}\right]} d x\) is :

86391 If \(f(x)\) is continuous at \(x=0\) and \(f(0)=2\), then what is the value of \(\lim _{x \rightarrow 0}\left[\frac{\int_{0}^{x} f(u) d u}{x}\right]\) ?

86392 Let \(f: \mathrm{R} \rightarrow \mathrm{R}\) be a differentiable function such that \(f(3)=3, f^{\prime}(3)=\frac{1}{2}\). Then the value of
\(\lim _{x \rightarrow 3} \int_{3}^{f(x)}\left[\frac{2 t^{3}}{x-3}\right] \text { is }\)

86393 If \(f: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) are continuous functions, then the value of the integral \(\int_{-\pi / 2}^{\pi / 2}(f(x)+f(-x))(g(x)-g(-x)) d x\) is

86394 If \(\int_{0}^{\pi n} f\left(\cos ^{2} x\right) \mathrm{d} x=k \int_{0}^{\pi} f\left(\cos ^{2} x\right) \mathrm{d} x\), then the value of \(k\) is

86395 If \([t]\) denotes the greatest integer \(\leq t\), then the value of \(\frac{3(e-1)}{e} \int_{1}^{2} x^{2} e^{[x]+\left[x^{3}\right]} d x\) is :

86391 If \(f(x)\) is continuous at \(x=0\) and \(f(0)=2\), then what is the value of \(\lim _{x \rightarrow 0}\left[\frac{\int_{0}^{x} f(u) d u}{x}\right]\) ?

86392 Let \(f: \mathrm{R} \rightarrow \mathrm{R}\) be a differentiable function such that \(f(3)=3, f^{\prime}(3)=\frac{1}{2}\). Then the value of
\(\lim _{x \rightarrow 3} \int_{3}^{f(x)}\left[\frac{2 t^{3}}{x-3}\right] \text { is }\)

86393 If \(f: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) are continuous functions, then the value of the integral \(\int_{-\pi / 2}^{\pi / 2}(f(x)+f(-x))(g(x)-g(-x)) d x\) is

86394 If \(\int_{0}^{\pi n} f\left(\cos ^{2} x\right) \mathrm{d} x=k \int_{0}^{\pi} f\left(\cos ^{2} x\right) \mathrm{d} x\), then the value of \(k\) is

86395 If \([t]\) denotes the greatest integer \(\leq t\), then the value of \(\frac{3(e-1)}{e} \int_{1}^{2} x^{2} e^{[x]+\left[x^{3}\right]} d x\) is :

86391 If \(f(x)\) is continuous at \(x=0\) and \(f(0)=2\), then what is the value of \(\lim _{x \rightarrow 0}\left[\frac{\int_{0}^{x} f(u) d u}{x}\right]\) ?

86392 Let \(f: \mathrm{R} \rightarrow \mathrm{R}\) be a differentiable function such that \(f(3)=3, f^{\prime}(3)=\frac{1}{2}\). Then the value of
\(\lim _{x \rightarrow 3} \int_{3}^{f(x)}\left[\frac{2 t^{3}}{x-3}\right] \text { is }\)

86393 If \(f: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) are continuous functions, then the value of the integral \(\int_{-\pi / 2}^{\pi / 2}(f(x)+f(-x))(g(x)-g(-x)) d x\) is

86394 If \(\int_{0}^{\pi n} f\left(\cos ^{2} x\right) \mathrm{d} x=k \int_{0}^{\pi} f\left(\cos ^{2} x\right) \mathrm{d} x\), then the value of \(k\) is

86395 If \([t]\) denotes the greatest integer \(\leq t\), then the value of \(\frac{3(e-1)}{e} \int_{1}^{2} x^{2} e^{[x]+\left[x^{3}\right]} d x\) is :

86391 If \(f(x)\) is continuous at \(x=0\) and \(f(0)=2\), then what is the value of \(\lim _{x \rightarrow 0}\left[\frac{\int_{0}^{x} f(u) d u}{x}\right]\) ?

86392 Let \(f: \mathrm{R} \rightarrow \mathrm{R}\) be a differentiable function such that \(f(3)=3, f^{\prime}(3)=\frac{1}{2}\). Then the value of
\(\lim _{x \rightarrow 3} \int_{3}^{f(x)}\left[\frac{2 t^{3}}{x-3}\right] \text { is }\)

86393 If \(f: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) are continuous functions, then the value of the integral \(\int_{-\pi / 2}^{\pi / 2}(f(x)+f(-x))(g(x)-g(-x)) d x\) is

86394 If \(\int_{0}^{\pi n} f\left(\cos ^{2} x\right) \mathrm{d} x=k \int_{0}^{\pi} f\left(\cos ^{2} x\right) \mathrm{d} x\), then the value of \(k\) is

86395 If \([t]\) denotes the greatest integer \(\leq t\), then the value of \(\frac{3(e-1)}{e} \int_{1}^{2} x^{2} e^{[x]+\left[x^{3}\right]} d x\) is :