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Integrals

Antiderivatives by substitution, parts and partial fractions.

Integration undoes differentiation. The solver shows which technique it uses (the power rule, a u-substitution, integration by parts, partial fractions) and the graph plots the antiderivative F against the integrand f, so you can see that F's slope is f.

ຕົວຢ່າງ​ທີ່​ໄດ້​ເຮັດ · integral of x*e^x dx

Calculus · Integrals

Integrate x·e^(x)

\int x e^{x}\, dx

ຂັ້ນ​ຕອນ​ຕໍ່​ຂັ້ນ​ຕອນ

  1. \int x e^{x}\, dx

    Find an antiderivative.

  2. u = x,\quad dv = e^{x}\, dx

    Integration by parts: ∫u dv = uv − ∫v du.

  3. du = 1\, dx,\quad v = e^{x}

    Differentiate u, integrate dv.

  4. \int x e^{x}\, dx = x e^{x} - \int e^{x}\, dx

    Apply the formula.

  5. \int e^{x}\, dx = e^{x}

    ∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).

  6. = x e^{x} - e^{x}

    Combine.

  7. F(x) = \left(x - 1\right) e^{x} + C

    Add the constant of integration.

ເປີດ​ເຜີຍ​ຄຳ​ຕອບ
\left(x - 1\right) e^{x} + C

ປັດຈຸບັນ​ທ່ານ ເລືອກ​ບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມ​ເອງ. ທຸກ​ບາດກ້າວ, ຮູບ, ຄໍາຕອບ​ຖືກ​ລົບ​ຈົນ​ກວ່າ​ເຈົ້າ​ຈະ​ຖາມ​ມັນ.

ຮັກສາ​ການ​ເຮັດວຽກ​ຂອງ​ທ່ານ​ໄວ້

ບັນຊີຟຣີ ເພີ່ມ​ຂໍ້​ມູນ​ໃສ່​ບົດຮຽນ​ທຸກ​ບົດ, ບັນທຶກ​ຂອງ​ສິ່ງທີ່​ທ່ານ​ໄດ້​ເຮັດ​ແລ້ວ, ບັນຫາ​ທີ່​ທ່ານ​ໄດ້​ແກ້​ໄຂ​ໄວ້​ໃນ​ບ່ອນ​ດຽວ ແລະ ຄູ​ສອນ​ທີ່​ທ່ານ​ສາມາດ​ຖາມ​ກ່ຽວກັບ​ໜ້າ​ນີ້. ຄະນິດສາດ​ເອງ​ແມ່ນ​ເປີດ​ໃຫ້​ທຸກ​ຄົນ, ບໍ່​ວ່າ​ຈະ​ໄດ້​ເຂົ້າ​ສູ່​ລະບົບ ຫຼື ບໍ່.

ລົງທະບຽນ ចូល

ຕົວ​ສະ​ແດງ​ທີ່​ໃຊ້​ຢູ່​ທີ່​ນີ້

ກົດ​ຕົວ​ອັກສອນ​ໃດ​ໜຶ່ງ​ເພື່ອ​ເບິ່ງ​ຄວາມ​ໝາຍ​ເຕັມ, ຮູບ ແລະ ຕົວອັກສອນ​ແຕ່ລະ​ຕົວ​ໃນ​ມັນ​ໝາຍ​ຄວາມ​ວ່າ​ແນວໃດ.

ວິທີການ: Integrals

  1. Find an antiderivative.
  2. Integration by parts: ∫u dv = uv − ∫v du.
  3. Differentiate u, integrate dv.
  4. Apply the formula.
  5. ∫ aᵘ du = aᵘ / ln a (for eˣ that is just eˣ).
  6. Combine.
  7. Add the constant of integration.

ຄໍາຖາມທີ່ຄົນຖາມ

What is a derivative in one sentence?

The slope of the graph at a point, the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate: the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function, a polynomial times an exponential, log or trig function.

ເພີ່ມເຕີມໃນ Calculus

LimitsDerivativesDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests