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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.
उदाहरण · integral of 2x/(x^2+1)
चरणद्वारा चरण
- \int \frac{2 x}{x^{2} + 1}\, dx
Find an antiderivative.
- \int \frac{2 x}{x^{2} + 1}\, dx = 2 \int \frac{x}{x^{2} + 1}\, dx
Pull the constant 2 out of the integral.
- \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.
- u = x^{2} + 1,\quad du = 2 x\, dx
Substitute u = x^{2} + 1.
- \int \frac{x}{x^{2} + 1}\, dx = \int \frac{1}{u}\, d_u
Rewrite the integral in terms of u.
- \int \frac{1}{u}\, d_u = \log{\left(u \right)}
∫ 1/u du = ln|u|.
- = \log{\left(x^{2} + 1 \right)}
Substitute back u = x^{2} + 1.
- F(x) = \log{\left(x^{2} + 1 \right)} + C
Add the constant of integration.
जवाफ प्रकट गर्नुहोस्
अब तपाईँ समस्या रोज्नुहोस्, वा टाइप गर्नुहोस् वा तपाईँको आफ्नै रेखाचित्र गर्नुहोस् । प्रत्येक चरण, एउटा तस्वीर, तपाईँले सोध्नु अघि उत्तर लुकेको छ ।
एक निःशुल्क खाता प्रत्येक पाठ मा नोट थप्छ, तपाईं के समाप्त गरेको छ को एक रेकर्ड, एक ठाउँमा आफ्नो समाधान समस्या, र एक शिक्षक तपाईं यो पृष्ठ बारेमा सोध्न सक्नुहुन्छ. गणित आफैलाई सबैलाई खुला छ, मा साइन इन वा छैन.
साइन अप गर्नुहोस् लगइनयहाँ प्रयोग गरिएको प्रतीक
पूर्ण परिभाषा, एउटा तस्वीर, र यो प्रत्येक अक्षर अर्थ के लागि कुनै पनि प्रतीक ट्याप गर्नुहोस्।
कसरी: Integration techniques: substitution, parts, partial fractions
- Look for a piece whose derivative is also present → substitution.
- A product of a polynomial with e^x, ln x, sin or cos → integration by parts, differentiating the polynomial.
- A rational function → divide if needed, then partial fractions.
- Powers of sin and cos → identities (sin² = (1 − cos 2x)/2) or a substitution.
- Check by differentiating your answer.
प्रश्नहरू मानिसहरूले सोध्छन्
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.
यसमा थप Calculus
LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationApplications of integration: area, volume, arc lengthInfinite series and convergence tests