Indefinite integrals Use a change of variables or Table 5.6 to...

Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.

∫ sin 𝓍 sec⁸ 𝓍 d𝓍

Accepted Answer

Welcome back, everyone. Determine the following indefinite integral. Integral of cosequent X multiplied by co-sequence the power of 5 of XD X. For this problem, we're going to rewrite cosequant of X in terms of sign. So let's recall that cosequant is equal to 1 divided by sign, right? So we get 1 divided by sine of X, and because we have a 5th power, we're going to include sign, raise to the power of 5 of X. Now, what we're going to do is introduce the U substitution. Let's suppose that U equals sin x. Then the u divided by D X is the derivative of sine which is cosine. And now we can show that the u equals cosine xD X. Which is exactly what we have in the numerator of the integral. So we can rewrite our integral as cosine xDX which is DU. Divided by sine raises the power of 5 of X, which is a u raises the power of 5. And now we can integrate using the power rule. This is you raise the power of -5DU. According to the power rule, the integral of you raised to the power of -5 is U raise to the power of -4 divided by -4 plus a constant of integration C. And now what we can do is take out the negative sign, we get negative. One divided by For you raise the power of 4 plus C, right? Essential, we have 4 in the denominator. You raise to the power of -4 can be rewritten as 1 divided by you raise to the power of 4. And now we can substitute back. We know that U equals sin x, so we get -1. Divided by 4, sine race to the power of 4 of X plus a constant of integration C. And at 1 divided by sine raises the power of 4 of X is basically the same cosic can traces the power of 4. So we can rewrite the answer as negative. Cosequant raise to the power of 4 of X divided by 4 + a constant of integration C. That would be our final answer, and thank you for watching.

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. ∫ sin 𝓍 sec⁸ 𝓍 d𝓍

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Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.

∫ sin 𝓍 sec⁸ 𝓍 d𝓍