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Questions and Answers
What is the integral $ rac{d}{du} igg( igg( rac{u^{n+1}}{n+1} igg) + C igg)$?
What is the integral $ rac{d}{du} igg( igg( rac{u^{n+1}}{n+1} igg) + C igg)$?
- ∫du
- ∫u^n du (correct)
- ∫(du/u)
- ∫kf(u)du
What is the result of the integral ∫kf(u)du?
What is the result of the integral ∫kf(u)du?
k∫f(u)du
What is the result of the integral ∫[f(u) ± g(u)]?
What is the result of the integral ∫[f(u) ± g(u)]?
∫f(u)du ± ∫g(u)du
What is the result of the integral ∫du?
What is the result of the integral ∫du?
What is the integral ∫u^n du?
What is the integral ∫u^n du?
What does ∫(du/u) equal?
What does ∫(du/u) equal?
What is the result of the integral ∫e^u du?
What is the result of the integral ∫e^u du?
What is the integral ∫sin(u) du?
What is the integral ∫sin(u) du?
What does ∫cos(u) du equal?
What does ∫cos(u) du equal?
What is the result of the integral ∫tan(u) du?
What is the result of the integral ∫tan(u) du?
What does ∫cot(u) du equal?
What does ∫cot(u) du equal?
What is the integral ∫sec(u) du?
What is the integral ∫sec(u) du?
What does ∫csc(u) du equal?
What does ∫csc(u) du equal?
What is the result of the integral ∫sec^2(u) du?
What is the result of the integral ∫sec^2(u) du?
What does ∫csc^2(u) du equal?
What does ∫csc^2(u) du equal?
What is the integral ∫sec(u)tan(u) du?
What is the integral ∫sec(u)tan(u) du?
What is the integral ∫csc(u)cot(u) du?
What is the integral ∫csc(u)cot(u) du?
What is the result of the integral ∫(du)/(√(a^2 - u^2))?
What is the result of the integral ∫(du)/(√(a^2 - u^2))?
What does ∫(du)/((a^2 - u^2)) equal?
What does ∫(du)/((a^2 - u^2)) equal?
What is the integral ∫(du)/(u√(u^2 - a^2))?
What is the integral ∫(du)/(u√(u^2 - a^2))?
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Study Notes
Basic Integration Formulas
- The integral of a constant multiplied by a function can be factored out: ∫kf(u)du = k∫f(u)du.
- The integral of the sum or difference of two functions is the sum or difference of their integrals: ∫[f(u) ± g(u)] = ∫f(u)du ± ∫g(u)du.
Indefinite Integrals
- The integral of ( du ) is ( u + C ).
- For any integer ( n \neq -1 ), the integral of ( u^n ) is (\frac{u^{n+1}}{n+1} + C).
Exponential and Logarithmic Functions
- The integral of (\frac{du}{u}) results in the natural logarithm: ( \ln|u| + C ).
- The integral of ( e^u ) is simply ( e^u + C ).
Trigonometric Functions
- The integral of ( \sin(u) ) yields: (-\cos(u) + C).
- The integral of ( \cos(u) ) results in: ( \sin(u) + C ).
- The integral of ( \tan(u) ) gives: (-\ln| \cos(u) | + C).
- The integral of ( \cot(u) ) results in: ( \ln| \sin(u) | + C).
- The integral of ( \sec(u) ) is expressed as: ( \ln| \sec(u) + \tan(u) | + C).
- The integral of ( \csc(u) ) leads to: (- \ln| \csc(u) + \cot(u) | + C).
- The integral of ( \sec^2(u) ) results in: ( \tan(u) + C).
- The integral of ( \csc^2(u) ) yields: (- \cot(u) + C).
- The integral of ( \sec(u) \tan(u) ) gives: ( \sec(u) + C).
- The integral of ( \csc(u) \cot(u) ) results in: (- \csc(u) + C).
Inverse Trigonometric Functions
- The integral of (\frac{du}{\sqrt{a^2 - u^2}}) produces ( \arcsin\left(\frac{u}{a}\right) + C ).
- The integral of (\frac{du}{a^2 - u^2}) results in ( \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C ).
- The integral of ( \frac{du}{u\sqrt{u^2 - a^2}} ) yields ( \frac{1}{a} \text{arcsec}\left(\frac{|u|}{a}\right) + C ).
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