maths.free › Calculus › Integrals

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.

작업 예제 · integral of x*e^x dx

Calculus · Integrals

Integrate x·e^(x)

\int x e^{x}\, dx

단계별로

  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.

답을 드러내세요
\left(x - 1\right) e^{x} + C

이제 너 문제를 선택하거나, 타이핑하거나, 자신만의 문제를 그려보세요. 모든 단계는 그림으로 표시되며, 답은 질문을 할 때까지 숨겨져 있습니다.

자신의 작업을 유지

무료 계정은 모든 수업에 노트를 추가, 당신이 완료 한 기록, 당신이 해결 한 문제를 한 곳에, 그리고 당신이이 페이지에 대해 물어 수업 지도자. 수학 자체는 모두에게 열려 있습니다, 로그인 여부.

가입하기 로그인

여기서 사용된 기호

기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.

어떻게: 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.

사람들이 묻는 질문

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.

에 더 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