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## Calculus 2

### Unit 1: Lesson 11

Indefinite integrals of common functions

# Common integrals review

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.C (LO)
,
FUN‑6.C.1 (EK)
,
FUN‑6.C.2 (EK)
Review the integration rules for all the common function types.

## Polynomials

integral, x, start superscript, n, end superscript, d, x, equals, start fraction, x, start superscript, n, plus, 1, end superscript, divided by, n, plus, 1, end fraction, plus, C

\begin{aligned} \displaystyle\int\sqrt[\Large m]{x^n}\,dx&=\displaystyle\int x^{^{\Large\frac{n}{m}}}\,dx \\\\ &=\dfrac{x^{^{\Large \frac{n}{m}+1}}}{\dfrac{n}{m}+1}+C \end{aligned}
Want to practice integrating polynomials and radicals? Check out these exercises:

## Trigonometric functions

integral, sine, left parenthesis, x, right parenthesis, d, x, equals, minus, cosine, left parenthesis, x, right parenthesis, plus, C
integral, cosine, left parenthesis, x, right parenthesis, d, x, equals, sine, left parenthesis, x, right parenthesis, plus, C
integral, \sec, squared, left parenthesis, x, right parenthesis, d, x, equals, tangent, left parenthesis, x, right parenthesis, plus, C
integral, \csc, squared, left parenthesis, x, right parenthesis, d, x, equals, minus, cotangent, left parenthesis, x, right parenthesis, plus, C
integral, \sec, left parenthesis, x, right parenthesis, tangent, left parenthesis, x, right parenthesis, d, x, equals, \sec, left parenthesis, x, right parenthesis, plus, C
integral, \csc, left parenthesis, x, right parenthesis, cotangent, left parenthesis, x, right parenthesis, d, x, equals, minus, \csc, left parenthesis, x, right parenthesis, plus, C
Want to practice integrating trigonometric functions? Check out these exercises:

## Exponential functions

integral, e, start superscript, x, end superscript, d, x, equals, e, start superscript, x, end superscript, plus, C
integral, a, start superscript, x, end superscript, d, x, equals, start fraction, a, start superscript, x, end superscript, divided by, natural log, left parenthesis, a, right parenthesis, end fraction, plus, C

## Integrals that are logarithmic functions

integral, start fraction, 1, divided by, x, end fraction, d, x, equals, natural log, vertical bar, x, vertical bar, plus, C
Want to learn more about integrating exponential functions and start fraction, 1, divided by, x, end fraction? Check out this video.
Want to practice integrating exponential functions and start fraction, 1, divided by, x, end fraction? Check out this exercise.

## Integrals that are inverse trigonometric functions

integral, start fraction, 1, divided by, square root of, a, squared, minus, x, squared, end square root, end fraction, d, x, equals, \arcsin, left parenthesis, start fraction, x, divided by, a, end fraction, right parenthesis, plus, C
integral, start fraction, 1, divided by, a, squared, plus, x, squared, end fraction, d, x, equals, start fraction, 1, divided by, a, end fraction, \arctan, left parenthesis, start fraction, x, divided by, a, end fraction, right parenthesis, plus, C

## Want to join the conversation?

• What is the difference between x^n dx and a^x dx? That is, why is one a polynomial and one an exponential function?
• In the second function, variable x is the exponent. That is why the second one is exponential function.
• Could someone please provide me with the proof for
integral of 1/(a^2 + x^2)
• 1/(a² + x²) = 1/(a²(1 + x²/a²)
Let x = a·tan(u)
dx = a·sec²(u) du

Therefore ∫1/(a² + x²) dx = ∫a·sec²(u) / a²(1 + tan²(u)) du = 1/a ∫sec²(u) / (1 + tan²(u)) du
But 1 + tan²(u) = sec²(u)
So ∫1/(a² + x²) dx = 1/a ∫ du = u/a + C

Substituting back for u (= arctan(x/a) ) gives
∫1/(a² + x²) dx = 1/a · arctan(x/a) + C

• Why is the integral of tan(x) not listed?
• Why isn't there an arccos integral function?
• probably because arcsin and arccos have almost identical derivatives. their derivatives are negatives of each other.
• Where is the video for the second function under "Exponential Functions" (the integral of a^x)? Where are the videos for the whole section of "Integrals that are inverse trigonometric functions" (the integral of 1/sqrt[(a^2)-(x^2)] and the integral of 1/sqrt[(a^2)+(x^2)]? I can't find these three functions mentioned anywhere in the videos.
• I agree some of the things are not explained, I can't find the videos
• All these integrals of trigonometric functions are really confusing for me. Do I have to just learn them by heart? Or is there some section I missed, where they are explained more intuitively?
• There are proofs out there for each trig function but it is much easier to just learn them by heart.
• in common integral review under exponeential functions how integration of ax is ax/ln{a} shoudnt it be like the polynomial example?
(1 vote)
• a^x is not a polynomial