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How to Interpret and Calculate the Indefinite Integral


Integrals are mainly split into two categories: Definite and indefinite integrals. The indefinite integral is the same as the anti-derivative.

A challenge in working with integrals is having to find the expression without determining the constant term. The symbol C that you add to the end of integrals represents this unknown constant of integration. To find the constant, you are dependent on having constraints, such that you can find C by substitution and equation solving.

Theory

The Indefinite Integral

f(x)dx = F(x) + C,F(x) = f(x)

Here, f(x) is called the integrand and C is called the constant of integration.

Example 1

Compute the integral ln x + ex + x3dx

ln x + ex + x3dx = x ln x x + ex + 1 4x4 + C

ln x + ex + x3dx = x ln x x + ex + 1 4x4 + C

Example 2

Compute the integral

3 cos(3x) 4 sin(2x)dx

3 cos(3x) 4 sin(2x)dx

= 3 cos(3x) 4 sin(2x)dx = 3 1 3 sin(3x) + 4 1 2 cos(2x) + C = sin(3x) + 2 cos(2x) + C

3 cos(3x) 4 sin(2x)dx = 3 1 3 sin(3x) + 4 1 2 cos(2x) + C = sin(3x) + 2 cos(2x) + C

Example 3

Compute the integral

sin(2x) + 3 cos(x) e5xdx

sin(2x) + 3 cos(x) e5xdx

= sin(2x) + 3 cos(x) e5xdx = 1 2 cos(2x) + 3 sin(x) 1 5e5x + C

sin(2x) + 3 cos(x) e5xdx = 1 2 cos(2x) + 3 sin(x) 1 5e5x + C

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How to Interpret and Calculate the Definite Integral
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