_-substitution: multiplying by a constant | AP Calculus AB | Khan Academy

TL;DR

Learn how to solve the indefinite integral of the square root of 7x + 9 using u-substitution.

Transcript

Let's take the indefinite integral of the square root of 7x plus 9 dx. So my first question to you is, is this going to be a good case for u-substitution? Well, when you look here, maybe the natural thing to set to be equal to u is 7x plus 9. But do I see its derivative anywhere over here? Well, let's see. If we set u to be equal to 7x plus 9, what... Read More

Key Insights

• 😄 U-substitution is a powerful technique for solving integrals when the derivative of the substitution variable appears in the integrand.
• ✖️ The multiplication and division of the integrand by scalars can simplify the integral and make it more amenable to manipulation.
• ✊ The antiderivative of u to the power of 1/2 is found by incrementing the power by 1 and multiplying by the reciprocal of the new power.

Questions & Answers

Q: What is the first step in determining whether u-substitution is suitable for solving an integral?

The first step is to check if the derivative of the function to be substituted, in this case, 7x + 9, is present in the integrand. If so, u-substitution can be applied.

Q: What is the purpose of multiplying and dividing the integrand by 7?

The multiplication and division by 7 is done to create a 7 in the integrand, which allows for the application of u-substitution. The scalar can be moved in and out of the integral easily.

Q: How is u-substitution utilized in solving the integral?

After setting u equal to 7x + 9, the derivative of u, 7, is found, allowing for the substitution of variables. The integral is then rewritten in terms of u and solved accordingly.

Q: What is the final step after solving the indefinite integral in terms of u?

The solution is presented as 2/21 times (7x + 9) to the power of 3/2, added with a constant term. This constant accounts for the arbitrary value that could not be determined solely through the indefinite integral.

Summary & Key Takeaways

• This video teaches how to solve the indefinite integral of the square root of 7x + 9 using u-substitution.

• The first step is to determine whether u-substitution is suitable by checking if the derivative of 7x + 9 is present in the integrand.

• By multiplying and dividing the integrand by 7, a scalar, the integral can be simplified and u-substitution becomes feasible.