Home > Calculus > Integration by Trigonometric Substitution

How do you integrate #(-6x-7x^2) dx#?

Answer 1

#-3x^2-7/3x^3+c#

Integrate each term using the #color(blue)"power rule for integration"#

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(int(ax^n)dx=a/(n+1)x^(n+1))color(white)(2/2)|)))#

#rArrint(-6x-7x^2)dx#

#=(-6)/2x^2-7/3x^3+c=-3x^2-7/3x^3+c#

where c is the constant of integration.

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

Answer 2

Isabella Ackerman

To integrate ( \int (-6x - 7x^2) , dx ):

Use the power rule of integration, which states that ( \int x^n , dx = \frac{x^{n+1}}{n+1} + C ), where ( n \neq -1 ).
Integrate each term separately: [ \begin{align*} &\int (-6x - 7x^2) , dx \ &= \int (-6x) , dx + \int (-7x^2) , dx \ &= -6 \int x , dx - 7 \int x^2 , dx \ \end{align*} ]
Apply the power rule to each term: [ \begin{align*} &=-6 \left( \frac{x^2}{2} \right) - 7 \left( \frac{x^3}{3} \right) + C \ &= -3x^2 - \frac{7}{3}x^3 + C \ \end{align*} ]

So, ( \int (-6x - 7x^2) , dx = -3x^2 - \frac{7}{3}x^3 + 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.