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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.

مثال عملي · integral of 2x/(x^2+1)

Calculus · Integrals

Integrate 2x/(x^2 + 1)

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

خطوة خطوة

  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.

كشفت الإجابة
\log{\left(x^{2} + 1 \right)} + C

الآن أنت اختار مشكلة، أو طبعها أو رسمها بنفسك، كل خطوة، صورة، الإجابة مخفية حتى تسأل.

إبقي عملك الخاص

إن الحساب المجاني يضيفه ملاحظات على كل درس، وسجل لما أنجزته، ومشاكلك التي حلتها في مكان واحد، ومعلم يمكنك أن تسأله عن هذه الصفحة. والرياضيات نفسها مفتوحة أمام الجميع، سواء كانوا مستخدمين أو غير مستخدمين.

التسجيل الدخول

الرموز المستخدمة هنا

لمس أي رمز للحصول على التعريف الكامل، والصورة، وما يعنيه كل حرف في ذلك.

كيف: 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.

أسئلة يطرحها الناس

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.

أكثر في Calculus

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