5.6-bo'lim: Aniq integralning ta'rifi

1 va 2 masalalar uchun integralni baholash uchun aniq integralning ta'rifidan foydalaning. \(x_{\,i}^*\) uchun har bir intervalning o'ng oxirgi nuqtasidan foydalaning.

1. \( \displaystyle \int_{1}^{4}{{2x + 3\,dx}}\) Yechim
2. \( \displaystyle \int_{0}^{1}{{6x\left( {x - 1} \right)\,dx}}\) Yechim
3. Baholash: \( \displaystyle \int_{4}^{4}{{\frac{{\cos \left( {{{\bf{e}}^{3x}} + {x^2}} \right)}}{{{x^4} + 1}}\,dx}}\) Yechim

4 va 5 masalalar uchun \( \displaystyle \int_{6}^{{11}}{{f\left( x \right)\,dx}} = - 7\) va \( \displaystyle \int_{6}^{{11}}{{g\left( x \right)\,dx}} = 24\) berilgan integralning qiymatini aniqlang.

1. \( \displaystyle \int_{{11}}^{6}{{9f\left( x \right)\,dx}}\) Yechim
2. \( \displaystyle \int_{6}^{{11}}{{6g\left( x \right) - 10f\left( x \right)\,dx}}\) Yechim
3. \( \displaystyle \int_{5}^{2}{{f\left( x \right)\,dx}} = 3\) va \( \displaystyle \int_{5}^{9}{{f\left( x \right)\,dx}} = 8\)ni hisobga olgan holda \( \displaystyle \int_{2}^{9}{{f\left( x \right)\,dx}}\) qiymatini aniqlang. Yechim
4. \( \displaystyle \int_{{ - 4}}^{{20}}{{f\left( x \right)\,dx}}\) qiymatini \( \displaystyle \int_{{ - 4}}^{0}{{f\left( x \right)\,dx}} = - 2\), \( \displaystyle \int_{{31}}^{0}{{f\left( x \right)\,dx}} = 19\) va \( \displaystyle \int_{{20}}^{{31}}{{f\left( x \right)\,dx}} = - 21\) hisobga olgan holda aniqlang. Yechim

8 va 9-masalalar uchun integralning grafigini chizing va integralning qiymatini aniqlash uchun aniq integralning maydon talqinidan foydalaning.

1. \( \displaystyle \int_{1}^{4}{{3x - 2\,dx}}\) Yechim
2. \( \displaystyle \int_{0}^{5}{{ - 4x\,dx}}\) Yechim

10 – 12 masalalar uchun quyidagi integrallarning har birini x ga nisbatan farqlang.

1. \( \displaystyle \int_{4}^{x}{{9{{\cos }^2}\left( {{t^2} - 6t + 1} \right)\,dt}}\) Yechim
2. \( \displaystyle \int_{7}^{{\sin \left( {6x} \right)}}{{\sqrt {{t^2} + 4} dt}}\) Yechim
3. \( \displaystyle \int_{{3{x^{\,2}}}}^{{ - 1}}{{\frac{{{{\bf{e}}^t} - 1}}{t}dt}}\) Yechim