7–64. Integration review Evaluate the following integrals.
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Question

7–64. Integration review Evaluate the following integrals.

7. ∫ dx / (3 - 5x)^4

Accepted Answer

Welcome back, everyone. Find the following in a growl. In this problem, we want to integrate DX divided by 2 + 6 X raise to the power of 4. So before we begin integration, we're going to simplify the expression slightly. We have integral DX divided by. What we can do is simply factor out the 2, right? So we get 2 multiplied by 1 + 3 X all raised to the power of 4. And now, based on the properties of exponents, we can distribute our exponents, so we get in a growl of D X divided by 2 to the power of 4, right? That's 16, multiplied by 1 + 3 X raise to the power of 4. And of course we can factor out the constant, so we get 1 divided by 16. Integral DX divided by. 1 + 3 x to the power of 4. From here, we're going to use the U substitution. So let's suppose that U equals 1 + 3 X. Our next step is to differentiate U with respect to X, so the u divided by D X is the derivative of 1 + 3 X, which is simply equal to 3, right, because the derivative of 3 X is 3, the derivative of 1 is 01 is a constant. So now what we can do is simply express DX and DX is equal to DU divided by 3 or 13 DU. So, so our integral I, let's label it as I. Can be rewritten as 1 divided by 16 in a roll of D X which is 1/3 DU. Divided by 1 + 3 X0 to the power of 4, so that's a U to the power of 4. Well done. And now we can factor out 1/3, 1/3 multiplied by 1/16 is 1 divided by 48. And our in a role is DU divided by U to the power of 4. And we can rewrite it with a negative exponent using the reciprocal rule. So we get 1 divided by 48 integral of U to the power of -4 DU. And from here we can apply the power rule, right? So we get 1 divided by 48 multiplied by. Us the power of -4 + 1 divided by -4 + 1. Plus the constant of integration C because this is an indefinite integral. Meaning we get -1 divided by 48. Multiplied by 1, divided by 3. Notice that -4 + 1 is -3. We have factored out the negative sign in front. And now we are multiplying by. You to the power of -3 or One divided by u cubed, we can apply the reciprocal rule. Plus C Now let's combine the constants. We get negative 1. Divided by 144 multiplied by U cube. So now we're substituting back for you, which is 1 + 3 X. And we have two cubes. Let's see Well done. So we have identified our final in a row. Our final answer would be -1 divided by 144, which is multiplied by 1 + 3 X cubed plus C. Thank you for watching.

7–64. Integration review Evaluate the following integrals.
7. ∫ dx / (3 - 5x)^4

Key Concepts

Integration Techniques

Substitution Method

Power Rule for Integration

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