Asosiy tarkibga o'tish

5.6-bo'lim : Aniq integralning ta'rifi

1-4-masalalar uchun integralni baholash uchun aniq integralning ta'rifidan foydalaning. \(x_{\,i}^*\) uchun har bir intervalning o'ng oxirgi nuqtasidan foydalaning.

  1. \( \displaystyle \int_{{ - 2}}^{1}{{7 - 4x\,dx}}\)
  2. \( \displaystyle \int_{0}^{2}{{3{x^2} + 4x\,dx}}\)
  3. \( \displaystyle \int_{{ - 1}}^{1}{{{{\left( {x - 3} \right)}^2}\,dx}}\)
  4. \( \displaystyle \int_{0}^{3}{{8{x^3} + 3x - 2\,dx}}\)
  5. Baholash: \( \displaystyle \int_{{ - 123}}^{{ - 123}}{{{{\cos }^6}\left( {2x} \right) - {{\sin }^8}\left( {4x} \right)\,dx}}\)

6 – 8 masalalar uchun \( \displaystyle \int_{{ - 2}}^{5}{{f\left( x \right)\,dx}} = 1\) va \( \displaystyle \int_{{ - 2}}^{5}{{g\left( x \right)\,dx}} = 8\) berilgan integralning qiymatini aniqlang.

  1. \( \displaystyle \int_{{ - 2}}^{5}{{ - 3g\left( x \right)\,dx}}\)
  2. \( \displaystyle \int_{{ - 2}}^{5}{{7f\left( x \right) - \frac{1}{4}g\left( x \right)\,dx}}\)
  3. \( \displaystyle \int_{5}^{{ - 2}}{{12g\left( x \right) - 3f\left( x \right)\,dx}}\)
  4. \( \displaystyle \int_{{13}}^{7}{{f\left( x \right)\,dx}} = - 9\) va \( \displaystyle \int_{{13}}^{{ - 1}}{{f\left( x \right)\,dx}} = - 12\)ni hisobga olgan holda \( \displaystyle \int_{7}^{{ - 1}}{{f\left( x \right)\,dx}}\) qiymatini aniqlang.
  5. \( \displaystyle \int_{0}^{5}{{f\left( x \right)\,dx}} = 10\) va \( \displaystyle \int_{5}^{6}{{f\left( x \right)\,dx}} = 3\) ni hisobga olgan holda \( \displaystyle \int_{0}^{6}{{4f\left( x \right)\,dx}}\) qiymatini aniqlang.
  6. \( \displaystyle \int_{2}^{{10}}{{f\left( x \right)\,dx}}\) qiymatini \( \displaystyle \int_{2}^{4}{{f\left( x \right)\,dx}} = - 1\), \( \displaystyle \int_{4}^{7}{{f\left( x \right)\,dx}} = 3\) va \( \displaystyle \int_{{10}}^{7}{{f\left( x \right)\,dx}} = - 8\) hisobga olgan holda aniqlang.
  7. \( \displaystyle \int_{2}^{{ - 5}}{{f\left( x \right)\,dx}} = 56\), \( \displaystyle \int_{7}^{2}{{f\left( x \right)\,dx}} = - 90\) va \( \displaystyle \int_{{ - 1}}^{7}{{f\left( x \right)\,dx}} = 45\) ekanligini hisobga olib, \( \displaystyle \int_{{ - 5}}^{{ - 1}}{{f\left( x \right)\,dx}}\) qiymatini aniqlang.

13 – 17 masalalar uchun integralning grafigini chizing va integralning qiymatini aniqlash uchun aniq integralning maydon talqinidan foydalaning.

  1. \( \displaystyle \int_{{ - 2}}^{1}{{12 - 5x\,dx}}\)
  2. \( \displaystyle \int_{0}^{4}{{\sqrt {16 - {x^2}} \,dx}}\)
  3. \( \displaystyle \int_{{ - 3}}^{3}{{5 - \sqrt {9 - {x^2}} \,dx}}\)
  4. \( \displaystyle \int_{{ - 1}}^{3}{{8x - 3\,dx}}\)
  5. \( \displaystyle \int_{1}^{6}{{\left| {x - 3} \right|\,dx}}\)

18 – 23 masalalar uchun quyidagi integrallarning har birini x ga nisbatan farqlang.

  1. \( \displaystyle \int_{{ - 8}}^{x}{{{{\bf{e}}^{\cos \left( t \right)}}\,dt}}\)
  2. \( \displaystyle \int_{2}^{{{x^{\,2}}}}{{\sqrt {\cos \left( t \right) + 3} \,dt}}\)
  3. \( \displaystyle \int_{0}^{{{{\bf{e}}^{3x}}}}{{\frac{1}{{{t^4} + {t^2} + 1}}dt}}\)
  4. \( \displaystyle \int_{{\sin \left( {9x} \right)}}^{8}{{\frac{{{{\bf{e}}^t}}}{{7t}}dt}}\)
  5. \( \displaystyle \int_{{{x^{\,3}}}}^{x}{{{{\cos }^4}\left( t \right) - {{\sin }^2}\left( t \right)\,dt}}\)
  6. \( \displaystyle \int_{{9x}}^{{\tan \left( x \right)}}{{\frac{{\cos \left( t \right) + 2}}{{\sin \left( t \right) + 4}}\,dt}}\)
  7. Cheklovni baholang: \(\mathop {\lim }\limits_{x \to 0} \frac{{\displaystyle \int_{0}^{x}{{{{\bf{e}}^{{t^2}}}\,dt}}}}{x}\)