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Integration techniques: substitution, parts, partial fractions

Recognising which technique an integrand wants.

Substitution when the integrand contains a function and its derivative; parts for products of unrelated functions; partial fractions for rational functions; trig identities for powers of sin and cos. Picture it: the antiderivative F plotted with f, F's slope is f everywhere. Think it: each technique is a derivative rule run backwards: chain, product, and the algebra of fractions.

Çözümlü örnek · integral of 2x/(x^2+1)

Calculus · Integrals

Integrate 2x/(x^2 + 1)

\int \frac{2 x}{x^{2} + 1}\, dx

Adım adım

  1. \int \frac{2 x}{x^{2} + 1}\, dx

    Find an antiderivative.

  2. \int \frac{2 x}{x^{2} + 1}\, dx = 2 \int \frac{x}{x^{2} + 1}\, dx

    Pull the constant 2 out of the integral.

  3. \int \frac{x}{x^{2} + 1}\, dx = \frac{1}{2} \int \frac{2 x}{x^{2} + 1}\, dx

    Pull the constant \frac{1}{2} out of the integral.

  4. u = x^{2} + 1,\quad du = 2 x\, dx

    Substitute u = x^{2} + 1.

  5. \int \frac{x}{x^{2} + 1}\, dx = \int \frac{1}{u}\, d_u

    Rewrite the integral in terms of u.

  6. \int \frac{1}{u}\, d_u = \log{\left(u \right)}

    ∫ 1/u du = ln|u|.

  7. = \log{\left(x^{2} + 1 \right)}

    Substitute back u = x^{2} + 1.

  8. F(x) = \log{\left(x^{2} + 1 \right)} + C

    Add the constant of integration.

Cevabı göster
\log{\left(x^{2} + 1 \right)} + C

Şimdi sen Bir problem seçin, ya da kendi probleminizi yazın ya da çizin. Her adım, bir resim, cevap sorulana kadar gizli.

Kendi çalışmalarınızı saklayın

Ücretsiz bir hesap her ders için notlar, bitirdiğiniz şeylerin kaydı, çözüldüğünüz problemlerin bir yerdeki kaydı ve bu sayfa hakkında soru sormak için bir öğretmen ekler. Matematik kendisi, giriş yapmış veya yapmamış herkese açıktır.

Kayıt Ol Giriş

Burada kullanılan sembolleri

Tüm tanım, resim ve her harfin anlamı için herhangi bir sembolü tıklayın.

Nasıl: Integration techniques: substitution, parts, partial fractions

  1. Look for a piece whose derivative is also present → substitution.
  2. A product of a polynomial with e^x, ln x, sin or cos → integration by parts, differentiating the polynomial.
  3. A rational function → divide if needed, then partial fractions.
  4. Powers of sin and cos → identities (sin² = (1 − cos 2x)/2) or a substitution.
  5. Check by differentiating your answer.

Sık sorulan sorular

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.

More in Calculus

LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationApplications of integration: area, volume, arc lengthInfinite series and convergence tests