86778 Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function defined by :
\(f(x)= \begin{cases}\max \left\{t^{3}-3 t\right\} ; x \leq 2 \\ t \leq x ; x \leq 2 \\ x^{2}+2 x-6 ; 2\lt x\lt 3 \\ {[x-3]+9} ; 3 \leq x \leq 5 \\ 2 x+1 ; x>5\end{cases}\)
Where \([t]\) is the greatest integer less than or equal to \(t\). Let \(\mathbf{m}\) be the number of points where
\(f\) is not differentiable and \(I=\int_{-2}^{2} f(x) d x\). Then the ordered pair \((m, I)\) is equal to :-

86779 For any integer \(n\), the integral
\(\int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}(2 n+1) x d x\) has the value

86782 \(\int \frac{\left(x^{2}+1\right) e^{x}}{(x+1)^{2}} d x=f(x) e^{x}+C\), Where \(C\) is a constant, then \(\frac{d^{3} f}{d x^{3}}\) at \(x=1\) is equal to :

86783 Let \(\alpha \in(0,1)\) and \(\beta=\log _{e}(1-\alpha)\). Let \(P_{n}(x)=x\) \(+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\ldots . .+\frac{x^{n}}{n}, x \in(0,1) . \quad\) Then the integral \(\int_{0}^{a} \frac{\mathbf{t}^{50}}{1-\mathbf{t}} \mathbf{d t}\) is equal to

86784 Let \(I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3 \ldots\) Then

86778 Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function defined by :
\(f(x)= \begin{cases}\max \left\{t^{3}-3 t\right\} ; x \leq 2 \\ t \leq x ; x \leq 2 \\ x^{2}+2 x-6 ; 2\lt x\lt 3 \\ {[x-3]+9} ; 3 \leq x \leq 5 \\ 2 x+1 ; x>5\end{cases}\)
Where \([t]\) is the greatest integer less than or equal to \(t\). Let \(\mathbf{m}\) be the number of points where
\(f\) is not differentiable and \(I=\int_{-2}^{2} f(x) d x\). Then the ordered pair \((m, I)\) is equal to :-

86779 For any integer \(n\), the integral
\(\int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}(2 n+1) x d x\) has the value

86782 \(\int \frac{\left(x^{2}+1\right) e^{x}}{(x+1)^{2}} d x=f(x) e^{x}+C\), Where \(C\) is a constant, then \(\frac{d^{3} f}{d x^{3}}\) at \(x=1\) is equal to :

86783 Let \(\alpha \in(0,1)\) and \(\beta=\log _{e}(1-\alpha)\). Let \(P_{n}(x)=x\) \(+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\ldots . .+\frac{x^{n}}{n}, x \in(0,1) . \quad\) Then the integral \(\int_{0}^{a} \frac{\mathbf{t}^{50}}{1-\mathbf{t}} \mathbf{d t}\) is equal to

86784 Let \(I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3 \ldots\) Then

86778 Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function defined by :
\(f(x)= \begin{cases}\max \left\{t^{3}-3 t\right\} ; x \leq 2 \\ t \leq x ; x \leq 2 \\ x^{2}+2 x-6 ; 2\lt x\lt 3 \\ {[x-3]+9} ; 3 \leq x \leq 5 \\ 2 x+1 ; x>5\end{cases}\)
Where \([t]\) is the greatest integer less than or equal to \(t\). Let \(\mathbf{m}\) be the number of points where
\(f\) is not differentiable and \(I=\int_{-2}^{2} f(x) d x\). Then the ordered pair \((m, I)\) is equal to :-

86779 For any integer \(n\), the integral
\(\int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}(2 n+1) x d x\) has the value

86782 \(\int \frac{\left(x^{2}+1\right) e^{x}}{(x+1)^{2}} d x=f(x) e^{x}+C\), Where \(C\) is a constant, then \(\frac{d^{3} f}{d x^{3}}\) at \(x=1\) is equal to :

86783 Let \(\alpha \in(0,1)\) and \(\beta=\log _{e}(1-\alpha)\). Let \(P_{n}(x)=x\) \(+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\ldots . .+\frac{x^{n}}{n}, x \in(0,1) . \quad\) Then the integral \(\int_{0}^{a} \frac{\mathbf{t}^{50}}{1-\mathbf{t}} \mathbf{d t}\) is equal to

86784 Let \(I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3 \ldots\) Then

86778 Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function defined by :
\(f(x)= \begin{cases}\max \left\{t^{3}-3 t\right\} ; x \leq 2 \\ t \leq x ; x \leq 2 \\ x^{2}+2 x-6 ; 2\lt x\lt 3 \\ {[x-3]+9} ; 3 \leq x \leq 5 \\ 2 x+1 ; x>5\end{cases}\)
Where \([t]\) is the greatest integer less than or equal to \(t\). Let \(\mathbf{m}\) be the number of points where
\(f\) is not differentiable and \(I=\int_{-2}^{2} f(x) d x\). Then the ordered pair \((m, I)\) is equal to :-

86779 For any integer \(n\), the integral
\(\int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}(2 n+1) x d x\) has the value

86782 \(\int \frac{\left(x^{2}+1\right) e^{x}}{(x+1)^{2}} d x=f(x) e^{x}+C\), Where \(C\) is a constant, then \(\frac{d^{3} f}{d x^{3}}\) at \(x=1\) is equal to :

86783 Let \(\alpha \in(0,1)\) and \(\beta=\log _{e}(1-\alpha)\). Let \(P_{n}(x)=x\) \(+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\ldots . .+\frac{x^{n}}{n}, x \in(0,1) . \quad\) Then the integral \(\int_{0}^{a} \frac{\mathbf{t}^{50}}{1-\mathbf{t}} \mathbf{d t}\) is equal to

86784 Let \(I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3 \ldots\) Then

86778 Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function defined by :
\(f(x)= \begin{cases}\max \left\{t^{3}-3 t\right\} ; x \leq 2 \\ t \leq x ; x \leq 2 \\ x^{2}+2 x-6 ; 2\lt x\lt 3 \\ {[x-3]+9} ; 3 \leq x \leq 5 \\ 2 x+1 ; x>5\end{cases}\)
Where \([t]\) is the greatest integer less than or equal to \(t\). Let \(\mathbf{m}\) be the number of points where
\(f\) is not differentiable and \(I=\int_{-2}^{2} f(x) d x\). Then the ordered pair \((m, I)\) is equal to :-

86779 For any integer \(n\), the integral
\(\int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}(2 n+1) x d x\) has the value

86782 \(\int \frac{\left(x^{2}+1\right) e^{x}}{(x+1)^{2}} d x=f(x) e^{x}+C\), Where \(C\) is a constant, then \(\frac{d^{3} f}{d x^{3}}\) at \(x=1\) is equal to :

86783 Let \(\alpha \in(0,1)\) and \(\beta=\log _{e}(1-\alpha)\). Let \(P_{n}(x)=x\) \(+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\ldots . .+\frac{x^{n}}{n}, x \in(0,1) . \quad\) Then the integral \(\int_{0}^{a} \frac{\mathbf{t}^{50}}{1-\mathbf{t}} \mathbf{d t}\) is equal to

86784 Let \(I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3 \ldots\) Then