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Verify and Confirm Basic Calculus Integration Formulas
Calculus (Mathematics)
High School Grade 12 / University First Year
Question Content
Verify and confirm the basic integration formulas listed: 1. $\\int kdx = kx + c$; 2. $\\int kF(x)dx = k\\int F(x)dx$; 3. $\\int [F(x)+g(x)]dx=\\int F(x)dx + \\int g(x)dx$; 4. $\\int x^n dx= \\frac{x^{n+1}}{n+1}+c$; 5. $\\int cosx dx=senx + c$; 6. $\\int senx dx=-cosx + c$; 7. $\\int sec^2x dx=tanx + c$; 8. $\\int secxtanx dx=secx + c$; 9. $\\int csc^2x dx=-cotx + c$; 10. $\\int csc(x)cotx dx=-cscx + c$
Correct Answer
All listed formulas are correct (note: $senx$ is the Spanish notation for $sinx$, the standard English notation for sine function)
Detailed Solution Steps
1
Step 1: Recognize that each formula is a standard basic integration rule, which can be verified by differentiating the right-hand side to match the integrand (the function being integrated).
2
Step 2: For example, verify formula 1: Differentiate $kx + c$ with respect to $x$, the result is $k$, which matches the integrand $k$.
3
Step 3: Verify formula 5: Differentiate $senx + c$ (or $sinx + c$) with respect to $x$, the result is $cosx$, which matches the integrand $cosx$.
4
Step 4: Repeat the differentiation verification for all remaining formulas: each derivative of the right-hand side will equal the integrand on the left-hand side, confirming all formulas are valid.
5
Step 5: Note that $senx$ is the Spanish-language notation for the sine function, equivalent to $sinx$ in standard English mathematical notation.
Knowledge Points Involved
1
Constant Rule for Integration
The integral of a constant $k$ with respect to $x$ is $kx + C$, where $C$ is the constant of integration. This rule comes from reversing the power rule for differentiation, and applies when integrating any constant value.
2
Linearity of Integration
This includes two sub-rules: the constant multiple rule ($\\int kF(x)dx = k\\int F(x)dx$) and the sum rule ($\\int [F(x)+g(x)]dx=\\int F(x)dx + \\int g(x)dx$). It states that integration distributes over addition and scalar multiplication, just like differentiation.
3
Power Rule for Integration
For any real number $n \\neq -1$, $\\int x^n dx= \\frac{x^{n+1}}{n+1}+C$. This rule is the reverse of the power rule for differentiation, used to integrate monomial functions of the form $x^n$.
4
Trigonometric Integration Rules
These are standard rules for integrating basic trigonometric functions and their products, such as $\\int cosx dx = sinx + C$, $\\int sec^2x dx=tanx + C$. Each rule is derived by reversing the corresponding trigonometric differentiation rule.
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