Methods of Integration

Name: Methods of Integration
Author: StemPy Lab

Some integrals do not yield to substitution alone. Learn the integrals of the special forms, including those with a quadratic under a root, split a fraction into partial fractions, and use integration by parts, which comes from the product rule. Choose the right method by looking at the shape of the integrand. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Integrals of special forms and the quadratic
Six integrals of the form $\int \dfrac{dx}{x^2 \pm a^2}$ and $\int \dfrac{dx}{\sqrt{a^2 \pm x^2}}$ are worth knowing by heart, together with three integrals of a square root. A quadratic $ax^2 + bx + c$ in a denominator or under a root is brought to one of these forms by completing the square. A fraction $\dfrac{px + q}{ax^2 + bx + c}$ is split into a part that gives a logarithm and a part that gives an inverse tangent.
1 simulation1 practice question1 Python program3 problems
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Lesson 2about 25 min
Partial fractions
A fraction whose bottom factorises can be split into simpler fractions, one for each factor. Distinct linear factors give $\dfrac{A}{x - a}$ , a repeated factor gives one term for each power, and an irreducible quadratic factor gives $\dfrac{Bx + C}{x^2 + p^2}$ . If the top has degree at least as large as the bottom, divide first. Each piece is then a standard integral.
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 3about 25 min
Integration by parts
Integration by parts reverses the product rule: $\int u\,v'\,dx = uv - \int u'\,v\,dx$ . Choose $u$ as the factor that becomes simpler when differentiated, and $v'$ as the factor you can integrate. It handles products such as $x\,e^x$ , $x\sin x$ and $x^2\ln x$ , and also $\ln x$ and $\tan^{-1}x$ on their own. The form $\int e^x\bigl[f(x) + f'(x)\bigr]dx = e^x f(x)$ is a shortcut that appears often.
1 simulation1 practice question1 Python program4 problems1 Python lab1 question set1 note
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Methods of Integration: Timed Test

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Methods of Integration

Course outline (3 lessons)

Integrals of special forms and the quadratic

Partial fractions

Integration by parts

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