What change of variables would you use for the integral ∫(4 - 7x)^(-6) dx?

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves a composite function (4 - 7x)^(-6). To simplify the integration, use a substitution method.

Step 2: Let u = 4 - 7x. This substitution simplifies the expression inside the integral. Compute the derivative of u with respect to x: du/dx = -7.

Step 3: Rearrange the derivative to express dx in terms of du: dx = du / (-7).

Step 4: Substitute u and dx into the integral. The integral becomes ∫u^(-6) * (du / -7).

Step 5: Factor out the constant -1/7 from the integral, leaving ∫u^(-6) du. This integral can now be solved using the power rule for integration.

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Change of Variables

Change of variables, also known as substitution, is a technique used in integration to simplify the integral by transforming it into a more manageable form. This involves substituting a new variable for an expression in the integral, which can make the integration process easier. The goal is to find a relationship between the original variable and the new variable that simplifies the integrand.

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