Indefinite Integral
Indefinite Integral
Indefinite Integral
Up to now we have been concerned with the branch of calculus called the differential calculus
involving the derivative. It involves the following problem: given a function, find its derivative (or
differential). We begin now with our study of the other branch of calculus called the integral calculus
pertaining to the definite (and indefinite) integral. It deals with the reverse problem: given the derivative,
find the function. We have the following definitions.
Note that a function has more than one antiderivative. In fact, if F (x) is an antiderivative of the
continuous function f (x) , then any other antiderivates of f (x) has the form F ( x) C for some
constant C. The family of all antiderivatives of f (x) is written as
integral sign f ( x) dx F ( x) C
constant of integration
integrand
integral
differential of x
dx x c
x n1
x dx n 1 C for all n 1
n
u n1
C for all n 1 ,
n
u du
n 1
where u is any function of x and du is the exact differential of u
af ( x)dx a f ( x)dx for any constant a
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EXERCISES: Evaluate the following integrals.
1. 5 dx 1
u
3
14. 2
3 du
x dx u2
4
2.
15. ( y 4 y 3 ) dy
5t dt
7
3.
3 2 y dy
2
5 u du
3 2 16.
4.
z 3 dz
4
3dx 17.
5. x2 dx
dy
18. x 5 2
6. y2
a x dx
3
19.
6dz
7. z4 20. 4 x 1 dx
2
x 4 x dx dx
3
8. 21. 2x 7
4 x 2 x 1 dx
4
3
9.
3 y 7 dy
2 x x dx
22.
2
10.
du
11. 6x 4x 1 dx
3 23. 4u a 3
2
15x 6x 2 dx
dy
4 2
12. 24.
3 y 2a
1
13. t dt
5 3 y 2 dy
3
t 25.
sec 4 x dx tan
2 2
2. 8. 3 y dy
1 cos ln x dx
3. cos 2 y dy 9. x
csc 2t dt sin x dx
2
4. 10.
cos4 x
5. csc5t cot 5t dt
1 1
6. sec 2 z tan 2 z dz
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Integration Yielding Logarithms
au
e du e C a du C, a > 0
u u u
and
ln a
e e e dt
2x t t 2
2. dx 7.
dy
e 1 e dx
2x 2x 3
ey
3. 8.
ze dz
4 z2
9.
ye dy
y2
4.
ye dy
3 y 2 1
10.
2 dx
x
5.
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