Home > Calculus > Integrals of Polynomial functions

What is the antiderivative of # (x+1)(2x-1)#?

Answer 1

Christopher Agresta

#int(x+1)(2x-1)dx=2/3x^3+1/2x^2-x+"c"#

Finding the integral of a function is equivalent to finding its antiderivative.

So we want to find #int(x+1)(2x-1)dx=int2x^2+x-1dx#

We apply the power rule to find this.

#intx^n dx=1/(n+1)x^(n+1)+"c"#

#rArrint2x^2+x-1dx=2/3x^3+1/2x^2-x+"c"#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Evelyn Agard

#=>(2/3x^2+1/2x-1)x + C#

where #C# is an arbitrary constant from the integration.

We start with:

#\int(x+1)(2x-1)dx#

We multiply the two binomials:

#=\int(2x^2+2x-x-1)dx#

#=\int(2x^2+x-1)dx#

We break up the integral into three separate integrals:

#=\int2x^2dx + \intxdx + \int -dx#

#=color(blue)(2intx^2dx) + color(orange)(intxdx)-color(green)(intdx)#

Now we integrate:

#=color(blue)(2(1/3x^3))+color(orange)(1/2x^2)-color(green)(x)+C#

Simplifying:

#=2/3x^3+1/2x^2-x+C#

#=(2/3x^2+1/2x-1)x + C#

where #C# is an arbitrary constant from the integration.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 3

Isaiah Allen

The antiderivative of (x+1)(2x-1) is ( \frac{1}{3}x^3 + \frac{1}{2}x^2 - x + C ), where C is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.