Integration Formulas 21B

Question 1

What is the result of $\int k \, dx$?

Accepted Answer

$ kx + C, \quad k 	ext{ any number} $

Question 2

What is the result of $\int x^n \, dx$?

Accepted Answer

$ \frac{x^{n+1}}{n+1} + C, \quad n 
eq -1 $

Question 3

What is the result of $\int \frac{dx}{x}$?

Accepted Answer

$ \ln |x| + C $

Question 4

What is the result of $\int e^x \, dx$?

Accepted Answer

$ e^x + C $

Question 5

What is the result of $\int a^x \, dx$?

Accepted Answer

$ \frac{a^x}{\ln a} + C, \quad (a > 0, a 
eq 1) $

Question 6

What is the result of $\int \sin x \, dx$?

Accepted Answer

$ -\cos x + C $

Question 7

What is the result of $\int \cos x \, dx$?

Accepted Answer

$ \sin x + C $

Question 8

What is the result of $\int \sec^2 x \, dx$?

Accepted Answer

$ 	an x + C $

Question 9

What is the result of $\int \csc^2 x \, dx$?

Accepted Answer

$ -\cot x + C $

Question 10

What is the result of $\int \sec x 	an x \, dx$?

Accepted Answer

$ \sec x + C $

Question 11

What is the result of $\int \csc x \cot x \, dx$?

Accepted Answer

$ -\csc x + C $

Question 12

What is the result of $\int 	an x \, dx$?

Accepted Answer

$ \ln |\sec x| + C $

Question 13

What is the result of $\int \cot x \, dx$?

Accepted Answer

$ \ln |\sin x| + C $

Question 14

What is the result of $\int \sinh x \, dx$?

Accepted Answer

$ \cosh x + C $

Question 15

What is the result of $\int \cosh x \, dx$?

Accepted Answer

$ \sinh x + C $

Question 16

What is the result of $\int \frac{dx}{\sqrt{a^2 - x^2}}$?

Accepted Answer

$ \arcsin \frac{x}{a} + C $

Question 17

What is the result of $\int \frac{dx}{a^2 + x^2}$?

Accepted Answer

$ \frac{1}{a} \arctan \frac{x}{a} + C $

Question 18

What is the result of $\int \frac{dx}{x\sqrt{x^2 - a^2}}$?

Accepted Answer

$ \frac{1}{a} \arcsec \left| \frac{x}{a} \right| + C $

Question 19

What is the result of $\int \frac{dx}{\sqrt{a^2 + x^2}}$?

Accepted Answer

$ \sinh^{-1} \frac{x}{a} + C, \quad (a > 0) $

Question 20

What is the result of $\int \frac{dx}{\sqrt{x^2 - a^2}}$?

Accepted Answer

$ \cosh^{-1} \frac{x}{a} + C, \quad (x > a > 0) $

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