Here are some solved questions
and answers for basic integration
rules:
1. Find the indefinite integral of
f(x) = 3x^2 + 4x - 7.
Solution: Using the power rule,
we can integrate each term
separately:
∫ 3x^2 dx = x^3 + C1 (where C1
is the constant of integration)
∫ 4x dx = 2x^2 + C2 (where C2 is
the constant of integration)
∫ -7 dx = -7x + C3 (where C3 is
the constant of integration)
Therefore, the indefinite integral
,of f(x) is:
∫ f(x) dx = x^3 + 2x^2 - 7x + C
where C is the constant of
integration.
,2. Find the definite integral of f(x)
= 4x + 1 from x = 0 to x = 2.
Solution: Using the sum and
difference rule, we can integrate
each term separately:
∫ 4x dx = 2x^2
∫ 1 dx = x
We can then evaluate the
definite integral as follows:
∫ 4x + 1 dx from x = 0 to x = 2
= [2x^2 + x] from x = 0 to x = 2
= (2(2)^2 + 2) - (2(0)^2 + 0)
=8+2
= 10
, Therefore, the definite integral of
f(x) from x = 0 to x = 2 is 10.
and answers for basic integration
rules:
1. Find the indefinite integral of
f(x) = 3x^2 + 4x - 7.
Solution: Using the power rule,
we can integrate each term
separately:
∫ 3x^2 dx = x^3 + C1 (where C1
is the constant of integration)
∫ 4x dx = 2x^2 + C2 (where C2 is
the constant of integration)
∫ -7 dx = -7x + C3 (where C3 is
the constant of integration)
Therefore, the indefinite integral
,of f(x) is:
∫ f(x) dx = x^3 + 2x^2 - 7x + C
where C is the constant of
integration.
,2. Find the definite integral of f(x)
= 4x + 1 from x = 0 to x = 2.
Solution: Using the sum and
difference rule, we can integrate
each term separately:
∫ 4x dx = 2x^2
∫ 1 dx = x
We can then evaluate the
definite integral as follows:
∫ 4x + 1 dx from x = 0 to x = 2
= [2x^2 + x] from x = 0 to x = 2
= (2(2)^2 + 2) - (2(0)^2 + 0)
=8+2
= 10
, Therefore, the definite integral of
f(x) from x = 0 to x = 2 is 10.
