76-81. Table of integrals Use a table of integrals to evaluate...

Question

76-81. Table of integrals Use a table of integrals to evaluate the following integrals.

76. ∫ x(2x + 3)⁵ dx

Accepted Answer

Welcome back, everyone. Evaluate the integral integral of T multiplied by 4 C minus 1 cubed DT. For this problem we're going to use substitution. Let's suppose that u equals 4 T minus 1. Then we also notice that we have T in our integran, right? So let's go for T. T is equal to U + 1. We have to add 1 to both sides and divide it by 4. Now, let's identify the relationship between DT and DU. If we identify the derivative of U with respect to T, we get 4. So we can conclude that the T is DU divided by 4, or simply 1/4 DU. So now we can rewrite our integral, integral of U + 1 divided by 4, multiplied by U. Cubed multiplied by the T, right? And DT is 14th DU. So now what we can do is simply factor out the constants. We have U + 1 divided by 4, and we also have 14th. So 14th multiplied by 1/4 is 1 divided by 16, and our integral contains a U cubed multiplied by U + 1 D U. Now let's go ahead and distribute this is 1/16 integral of you raise the power of 4 plus U cubed D U. And now let's integrate, let's integrate. We got 1/16 in the integral of U to the power of 4 is the U to the power of 5 divided by 5 using the power rule. And the integral of you cubed is you raise the power of 4 divided by 4 using the same power rule. And then we're going to add the constant of integration C because we have an indefinite integral. This is equal to 1/16th, and now we're going to find the common denominator before we substitute 40 minus 1 back. So er we're going to have 4 EU raise to the power of 5 plus 5, EU raise to the power of 4. Divided by 2. Plus a constant of integration C. Now we can remove currencies because we have a fraction. And we can also factor out you raise to the power of 4. So we get you raise to the power of 4 divided by 320. In Cs, we are going to have 4 U + 5 plus a constant of integration C. And then we can replace you with 4T minus 1. So we got 4T minus 1, raise to the power of 4. That's our first term. Then we get 4U, which is 16 C minus 4. Plus 5 all divided by 320 plus a constant of integration C and now -4 + 5 is 1 so we can replace it with 1 and we get our final answer. Thank you for watching.

76-81. Table of integrals Use a table of integrals to evaluate the following integrals.
76. ∫ x(2x + 3)⁵ dx

Key Concepts

Integration

Table of Integrals

Polynomial Expansion

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