Evaluating Definite Integrals

Evaluate the in...

Question

Evaluating Definite Integrals

Evaluate the integrals in Exercises 47–68.

∫₋₁¹ (3x² - 4x + 7)dx

Accepted Answer

Welcome back everyone. Evaluate the integral from -2 up to 1 of 4 x 2 minus 6 x plus 5 dx. A 36 B 6 T4 C 32. In the 18. So for this problem, let's go ahead and rewrite the integral. We are integrating from -2 up to 1. The function is 4 x 2 minus 6 x + 5dx. So what we can do is simply apply the power rule by factoring out each constant. For the first part we're going to factor out 4 and apply the power rule, so we get x to the power of 2 + 1 divided by 2 + 1 minus. We have 6 x to the power of 1, so the integral would be x 1 + 1 divided by 1 + 1. And for the final value, we have the integral of 5DX, right? So that would be plus 5 and the integral of DX is X. Well then, so now we want to evaluate the result from -2 up to 1 using the fundamental theorem of calculus part 2. But before we do that, let's just simplify the function slightly. So we have 4/3. X cubed. -6 divided by 2 gives us 3 x 2 + 5 x. We want to evaluate this function from -2 up to 1. Let's apply the limits of integration. At 1 we get 4/3 multiplied by 1 cubed minus 3 multiplied by 12 + 5 multiplied by 1. And we want to subtract the value at -2, which would be 4/3 multiplied by -2 cubed minus 3 multiplied by -22 + 5 multiplied by -2. And now let's perform. The calculations. Starting with the first part. We get 4/3 multiplied by 1, which is 4/3. -3 multiplied by 1, that's -3 + 5 multiplied by 1, that's 5 minus, we're going to distribute 4/3 multiplied by -2 cubed is 8. So -8 multiplied by 4 is -32. We get 32 divided by 3 and minus -32 divided by 3 is 32 divided by 3, right? Then we have -3 multiplied by -22, -22 is 4. And -3 multiplied by 4 is -12, and -5-12 is 12. And finally, -10, 5 multiplied by 2 is -10. 10 is 10. Now, let's combine everything. 4/3 plus 32/3 is 36/3, which is basically 12. -3 + 5 + 12 + 10 So we got 12 minus 3, which is 99 plus 5 is 1414 plus 12 is 26, plus 10, we get 36, which corresponds to the answer of choice A. Thank you for watching.

Evaluating Definite IntegralsEvaluate the integrals in Exercises 47–68.∫₋₁¹ (3x² - 4x + 7)dx

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Evaluating Definite Integrals

Evaluate the integrals in Exercises 47–68.

∫₋₁¹ (3x² - 4x + 7)dx