How do you find the indefinite integral of #int (-24x^5-10x) dx#?

Question

Emma Aldridge · Accepted Answer

#=-4x^6-5x^2+C# Knowing the method of polynomial integration that says: #color(red)(intx^ndx=x^(n+1)/(n+1))# #int(-24x^5-10x)dx# #=int(-24x^5)dx-int(10xdx)# #=(-24/color(red)6)x^color(red)(5+1)-10/color(red)2(x^(color(red)1+1))# #=-4x^6-5x^2+C#

Paisley Johnson · Answer

undefined To find the indefinite integral of $ \int (-24x^5 - 10x) \, dx $:

1. Use the power rule for integration.
2. Add 1 to the exponent and divide by the new exponent.
3. Apply the constant multiple rule for integration.

The result is:
$$ -\frac{24}{6}x^6 - \frac{10}{2}x^2 + C $$

Simplify to obtain the final result:
$$ -4x^6 - 5x^2 + C $$

Isabella Ackerman · Answer

undefined To find the indefinite integral of $ \int (-24x^5 - 10x) \, dx $, you can use the power rule for integration. This rule states that for a term of the form $ ax^n $, where $ a $ is a constant and $ n $ is any real number except for -1, the indefinite integral is given by $ \frac{a}{n+1}x^{n+1} + C $, where $ C $ is the constant of integration.

Applying this rule to each term separately:

For $ -24x^5 $, the integral becomes $ \frac{-24}{5+1}x^{5+1} = -4x^6 $.

For $ -10x $, the integral becomes $ \frac{-10}{1+1}x^{1+1} = -5x^2 $.

Adding these two results together, the indefinite integral of $ \int (-24x^5 - 10x) \, dx $ is $ -4x^6 - 5x^2 + C $, where $ C $ is the constant of integration.