7–40. Table look-up integrals Use a table of integrals to eval...

Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
8. ∫ sin 3x cos 2x dx

8. ∫ sin 3x cos 2x dx

Accepted Answer

Welcome back, everyone. Evaluate the integral of sine of 4 X cosine of 3XDX using trigonometric identities. For this problem we're going to use the product to some identity. Let's recall that if we have sine of alpha multiplied by cosine of beta, this product can be rewritten as a sum. That'd be 1/2 in sign of alpha plus beta plus of alpha minus beta. Now, our alpha is 4X, that's our first angle, and our beta is 3X. So we can basically show that sign of 4 X. Cosine of 3 X is equivalent to 1/2 in sin of alpha plus beta, so that would be 7 X. Plus sine of alpha minus beta, so 4 x minus 3 X, which is X. And now we are basically integrating one half multiplied by. The sum between. Sign of 7 X. And sign of X. D X We can also factor out the constant 1/2, and then we can split our integral into two parts, so that'd be 1/2 in the integral of sine of 7 XDX plus the integral of sine of XD X. And now we are ready to evaluate each integral, so we got 1/2. Let's recall that the integral of sine is equal to negative cosine, so we get negative cosine of 7 X, right, using the same angle. But according to the chain rule, we also have to divide by 7, right? So when we integrate, essentially, we have to recall that we are dividing by the slope of the linear term if we have a linear function, and indeed we have a linear function 7X. So, basically, negative cosine of 7 X must be divided by 7. So we get negative 1/7, cosine of 7 X, and we can check that by differentiation. According to the chain rule, as we mentioned. The derivative of cosine of 7 X is negative sign of 7 X and the negative sign gets canceled out, and then we're multiplying by the derivative of 7 X, which is 7, right? And 7 divided by 7 is 1, so we indeed get sin of 7 X. And now for the second integral, we are integrating sine xDX which is negative cosine of X. And we want to add a constant of integration because we are evaluating an indefinite integral. And now let's distribute. We get -1/14th cosine of 7 X. And then we are subtracting one half cosine of x and adding a constant of integration C. So that'd be our final answer for this problem. Thank you for watching.

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
8. ∫ sin 3x cos 2x dx

Key Concepts

Trigonometric Identities

Integration Techniques

Completing the Square

Watch next

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.8. ∫ sin 3x cos 2x dx

Key Concepts

Trigonometric Identities

Integration Techniques

Completing the Square

Watch next

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
8. ∫ sin 3x cos 2x dx