Definite Integral: Property

It says
the given equation is true for all x.

So, if x = 1,
the equation is true.

So put 1 into the equation.
Then ∫₁¹ f(t) dt = 1² + a⋅1.

The upper limit and the lower limit
are the same: 1.

So the left side is 0.

Simplify the right side.
1² + a⋅1 = 1 + a

So 0 = 1 + a.

Solve this equation.

Then a = -1.

So
a = -1
is the answer.

A definite integral is a value,
not a function.

So you can change
the variable of a definite integral.

+∫₁³ 2y² dy = +∫₁³ 2x² dx

-∫₁³ z dz = -∫₁³ x dx

These three integrals all have
the same upper limit (1) and the lower limit (3).

So ∫₁³ (x² + x - 1) dx + ∫₁³ 2x² dx - ∫₁³ x dx
= ∫₁³ (x² + x - 1 + 2x² - x) dx.

x² + 2x² = 3x²

Cancel +x and -x.

Then ∫₁³ (3x² - 1) dx.

Solve the integral.

Definite Integral: How to Solve

The integral of (3x² - 1) is
3⋅[1/3]x³ - x.

Integral of a Polynomial

3⋅[1/3]x³ - x = x³ - x

Put 3 and 1
into x³ - x.

Then 3³ - 3 - (1³ - 1).

3³ = 27

1³ = 1

27 - 3 = 24

1 - 1 = 0

24 - 0 = 24

So 24 is the answer.

The first integral is
from 0 to 1.

The second integral is
from 1 to 4.

And the inner functions of both integrals, (2x + 1),
are the same.

Then (given) = ∫₀⁴ f(x) dx.

Solve the integral.

The integral of (2x + 1) is
2⋅[1/2]x² + x.

2⋅[1/2]x² + x = x² + x

Put 4 and 0
into x² + x.

Then 4² + 4 - (0² + 0).

4² = 16

-(0² + 0) = 0

16 + 4 = 20

So 20 is the answer.

Write ∫₀⁸ 5x⁴ dx and -.

∫₂⁸ 5x⁴ dx = -∫₈² 5x⁴ dx

So (given) = ∫₀⁸ 5x⁴ - (-∫₈² 5x⁴ dx).

-(-∫₈² 5x⁴ dx) = +∫₈² 5x⁴ dx

The first integral is
from 0 to 8.

The second integral is
from 8 to 2.

And the inner functions of both integrals, 5x⁴,
are the same.

So (given) = ∫₀² 5x⁴ dx

Property 3 is used.

Solve the integral.

The integral of 5x⁴ is
5⋅[1/5]x⁵.

[1/5]x⁵ = x⁵

Put 2 and 0
into x⁵.

Then 2⁵ - 0⁵.

2⁵ = 32

So 32 is the answer.