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Evaluate the integral. \int x^{9}\cos(x^{5})\,dx

Question:

Evaluate the integral.

{eq}\displaystyle \int x^{9}\cos(x^{5})\,dx {/eq}

Integrals :

We have an indefinite integral that has a cosine term and a polynomial term. We will first use the substitution and then we will apply integration by parts. In the end, we will apply the constant of integration.

Answer and Explanation:

$$\int x^{9}\cos(x^{5})\,dx $$

We will do the following substitution:

$$x^5=u\\ =\int \frac{u\cos \left(u\right)}{5}du\\ =\frac{1}{5}\cdot \int \:u\cos \left(u\right)du\\ $$

Now we will use the integration by parts:

$$u=u,~v'=\cos u\\ =\frac{1}{5}\left(u\sin \left(u\right)-\int \sin \left(u\right)du\right)\\ =\frac{1}{5}\left(u\sin \left(u\right)-\left(-\cos \left(u\right)\right)\right)\\ $$

Putting the value of u back

$$=\frac{1}{5}\left(x^5\sin \left(x^5\right)+\cos \left(x^5\right)\right)+C $$


Learn more about this topic:

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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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