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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.

Иштөө мисалы: integral of x*e^x dx

Calculus · Integrals

Integrate x·e^(x)

\int x e^{x}\, dx

Аткаруу

  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.

Жауап
\left(x - 1\right) e^{x} + C

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Кээ бирлери Calculus

LimitsDerivativesDefinite integralsTaylor seriesSeries and sumsMaxima and minima