TÌM NGUYÊN HÀM NHANH NHẤT - ĐẦY ĐỦ NHẤT | TOÁN 12 SGK MỚI | Thầy Phạm Tuấn

Summary

The lesson explains how to find antiderivatives by reversing differentiation. It applies the main formula families to algebraic, exponential, trigonometric, and rational functions, then shows how antiderivatives can recover quantities from their rates of change.

1. Meaning and basic properties

A function (F) is an antiderivative of (f) if (F’(x)=f(x)). If (F) is one antiderivative, then all antiderivatives are (F(x)+C), where (C) is an arbitrary constant:

[ \int f(x)\,dx=F(x)+C. ]

A function has one derivative, but generally a whole family of antiderivatives that differ by constants.

Antiderivatives are linear:

[ \int(f+g)\,dx=\int f(x)\,dx+\int g(x)\,dx, \qquad \int kf(x)\,dx=k\int f(x)\,dx. ]

Sums and differences can be handled term by term. There is no general product or quotient rule for antiderivatives, so simplify or transform the expression first.

2. Core formulas

Constants and powers

[ \int a\,dx=ax+C, ]

[ \int x^\alpha\,dx=\frac{x^{\alpha+1}}{\alpha+1}+C \qquad(\alpha\ne -1), ]

[ \int\frac{1}{x}\,dx=\ln|x|+C. ]

The power formula also covers negative and fractional exponents. For example, rewrite (1/x^2) as (x^{-2}), or (\sqrt{x}) as (x^{1/2}), before applying it. When products contain powers of (x), combine the powers first.

Linear expressions inside a function

For (a\ne0),

[ \int (ax+b)^\alpha\,dx =\frac{(ax+b)^{\alpha+1}}{a(\alpha+1)}+C \qquad(\alpha\ne-1), ]

[ \int\frac{1}{ax+b}\,dx=\frac{1}{a}\ln|ax+b|+C. ]

The factor (1/a) accounts for the derivative of the inner expression (ax+b).

Exponential functions

[ \int e^x\,dx=e^x+C, \qquad \int e^{ax+b}\,dx=\frac{1}{a}e^{ax+b}+C, ]

and, for (u>0) with (u\ne1),

[ \int u^x\,dx=\frac{u^x}{\ln u}+C, \qquad \int u^{ax+b}\,dx=\frac{u^{ax+b}}{a\ln u}+C. ]

For products or quotients of exponentials, use exponent rules to combine them into a single exponential where possible.

Trigonometric functions

[ \int\sin x\,dx=-\cos x+C, \qquad \int\cos x\,dx=\sin x+C, ]

[ \int\frac{1}{\sin^2x}\,dx=-\cot x+C, \qquad \int\frac{1}{\cos^2x}\,dx=\tan x+C. ]

For arguments of the form (ax+b), include the corresponding factor (1/a).

3. A practical strategy for choosing a formula

The lesson emphasizes this general approach:

1. Simplify and rewrite the integrand so it resembles a known formula.
2. Split sums and differences and keep constant factors outside.
3. Use algebraic rules for powers, exponentials, or trigonometric expressions.
4. Apply the matching formula, including any required factor from a linear inner expression.
5. Add (C) and check the result by differentiating if needed.

4. Trigonometric transformations

When an expression does not directly match a standard formula, use identities to transform it first. The lesson reviews identities such as:

[ \sin^2x+\cos^2x=1, \qquad \sec^2x=1+\tan^2x, \qquad \csc^2x=1+\cot^2x, ]

[ \sin^2x=\frac{1-\cos 2x}{2}, \qquad \cos^2x=\frac{1+\cos 2x}{2}, ]

and the product-to-sum identity

[ \sin A\cos B=\tfrac12[\sin(A+B)+\sin(A-B)]. ]

These identities can reduce powers or turn products into sums, after which the terms can be integrated separately.

5. Differential method for composite functions

For a composite expression, identify an inner function (u) and look for its derivative (du) elsewhere in the integrand. Rewrite the integral in terms of (u), adjusting by a constant if the derivative is present only up to a factor.

Examples include:

• (\int x^2\cos(x^3)\,dx): the factor (x^2) supplies the derivative of (x^3), up to a constant.
• (\int \sin^4x\cos x\,dx): use (u=\sin x), with (\cos x\,dx=du).

The same idea can handle expressions such as (\tan x) by rewriting them as (\sin x/\cos x) and using the derivative of the denominator. The key is to recognize a function and its derivative together.

6. Rational functions

The lesson presents a stepwise procedure for rational integrands:

1. Compare polynomial degrees. If the numerator’s degree is at least the denominator’s, perform polynomial long division first.
2. Use direct formulas for simple linear denominators, such as (1/(ax+b)), or squared linear denominators, such as (1/(ax+b)^2).
3. For a linear numerator over a squared linear denominator, rewrite the numerator by adding and subtracting a suitable term so the expression separates into standard forms.
4. For distinct linear factors, use partial fractions. Set up constants over each factor, find the constants by matching coefficients or substituting the roots, then integrate each term.
5. For repeated factors, include terms for the successive powers of the repeated factor in the decomposition.

The aim is to break a rational expression into pieces that match formulas already learned.

7. Applications: recovering a quantity from its rate

Antiderivatives turn a rate of change into an accumulated quantity. The constant (C) is determined from an initial value or another condition.

Examples discussed include:

• Motion: Integrating velocity gives position or height. Initial position determines (C), and a stopping time can be found by setting velocity to zero.
• Plant growth: Integrating a growth rate gives plant height. The sign of the rate indicates growth or decline; maxima can be found from critical points and endpoint values.
• Event attendance: Integrating the rate of change in attendance gives the number of attendees. A known attendance value fixes (C), and maxima are checked over the stated time interval.
• Bacteria and other changing quantities: Integrate the given rate and use a known value to determine the integration constant.
• Water pumping: Integrate the pumping rate to obtain volume, then use measurements and an initial condition to solve for unknown parameters.
• Drug concentration: A model of the form (y’=ky) leads to an exponential expression (y=Ce^{kt}). Measurements can be used to determine the constants.

Speakers and sources featured

• Thầy Phạm Tuấn — the teacher and primary speaker.
• Students in the classroom — occasionally respond or are addressed, but the subtitles do not consistently identify distinct student speakers.
• External sources: None clearly featured in the subtitles.