Standard Integrals

Question 1

∫ (x^n) dx = ?

Answer 1

[(x^(n+1)) / (n+1)] + c

c is the arbitrary constant

Question 2

∫ (sin(x)) dx = ?

Answer 2

-cos(x) + c

Note these difference between the integrals and derivatives of sine and cosine

Question 3

∫ (sin(ax)) dx = ?

Answer 3

(-1/a * cos(ax)) + c

Note these difference between the integrals and derivatives of sine and cosine

Question 4

∫ (cos(x)) dx = ?

Answer 4

sin(x) + c

Question 5

∫ (cos(ax)) dx = ?

Answer 5

(1/a * sin(ax)) + c

Question 6

What general rule can be brought out for integrating trigonometric functions?

Answer 6

when integrating trig functions, differentiate the inner function and divide the integral of the outer function by this differential, this gives you the integral of the 2 functions together

Question 7

∫ (cos(ax^3)) dx = ?

Answer 7

[(1/(3ax^2)) * sin(ax^3)] + c

Question 8

∫ (1/x) dx = ?

where x > 0

Answer 8

ln|x| + c

Note the absolute value as there cannot be a natural log of a negative number

Question 9

∫ (e^x) dx = ?

Answer 9

(e^x) + c

the exponenetial function is the only function that integrates and differentiates to itself

Question 10

∫ (e^ax) dx = ?

where a is a constant and is not equal to 0

Answer 10

[(e^ax)/a] + c

Question 11

What general rule can be brought out for integrating exponential functions?

Answer 11

when integrating exponential functions, differentiate the power to which the function is raised and divide the function by this differential, do not forget to add c

Question 12

∫ (a^x) dx = ?

where a is a constant and is not equal to 0

Answer 12

[(a^x)/ln(a)] + c