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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:
- "Calculus: Early Transcendentals" by James Stewart
Video:
- 作者:现代数学启蒙
- 链接:https://www.math1234567.com/alevel006
- 声明:本文采用 CC BY-NC-SA 4.0 许可协议，转载请注明出处。
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