A Level 9709: Integration | 现代数学启蒙

slug

type

status

Topic: Techniques of Integration

In calculus, integration is the process of finding the integral of a function. It is the reverse operation of differentiation and is used to find the area under a curve, the length of a curve, the volume of a solid, and many other applications in mathematics and physics.

Example:

Find the integral of ∫(2x + 5) dx

To find the integral of ∫(2x + 5) dx, we can use the power rule of integration. The power rule states that the integral of x^n is (x^(n+1))/(n+1), where n is any real number except -1.

To apply the power rule, we distribute the integral to each term in the function: ∫(2x + 5) dx = ∫2x dx + ∫5 dx

Integrating each term separately, we get: = 2∫x dx + 5∫dx = 2(x^2/2) + 5x + C = x^2 + 5x + C

Therefore, the integral of ∫(2x + 5) dx is x^2 + 5x + C, where C is the constant of integration.

Exercises:

Find the integral of ∫(3x^2 + 4x - 1) dx

Evaluate ∫(sin x + cos x) dx

Calculate ∫(e^x + ln x) dx

Answers for the exercises above:

answer 1

To find the integral of ∫(3x^2 + 4x - 1) dx, we can use the power rule of integration. Applying the rule to each term, we get: = ∫3x^2 dx + ∫4x dx - ∫1 dx = 3(x^3/3) + 4(x^2/2) - x + C = x^3 + 2x^2 - x + C

Therefore, the integral of ∫(3x^2 + 4x - 1) dx is x^3 + 2x^2 - x + C, where C is the constant of integration.

answer2

To evaluate ∫(sin x + cos x) dx, we can integrate each term separately: = ∫sin x dx + ∫cos x dx = -cos x + sin x + C

Therefore, the integral of ∫(sin x + cos x) dx is -cos x + sin x + C, where C is the constant of integration.

answer3

To calculate ∫(e^x + ln x) dx, we integrate each term separately: = ∫e^x dx + ∫ln x dx = e^x + x(ln x - 1) + C

Therefore, the integral of ∫(e^x + ln x) dx is e^x + x(ln x - 1) + C, where C is the constant of integration.

Formula

Other Resources:

Integration Techniques Cheat Sheet

"Calculus: Early Transcendentals" by James Stewart

https://www.geogebra.org/m/fhmxtyw9