"Evaluate the following integral. ∫x3sin⁡(x4)cos5(x4)dx" was answered by our community

High School
Calculus and Analysis
Calculus 2
Applications of integrals

zagonek34

Answered question

2021-12-29

Evaluate the following integral.

\int x^{3} \sin (x^{4}) \cos^{5} (x^{4}) d x

Answer & Explanation

Joseph Fair

Beginner2021-12-30Added 34 answers

Step 1
Consider the given integral

\int x^{3} \sin (x^{4}) \cos^{5} (x^{4}) d x

Step 2
Use substitution method to evaluate the integral
Let

u = \cos (x^{4})

Then

d u = - 4 x^{3} \sin (x^{4}) d x

\Rightarrow x^{3} \sin (x^{4}) d x = - \frac{1}{4} d u

Step 3
Then the integral becomes

\int x^{3} \sin (x^{4}) \cos^{5} (x^{4}) d x = \int u^{5} (- \frac{1}{4}) d u

= - \frac{1}{4} \int u^{5} d u

= - \frac{1}{4} [\frac{u^{6}}{6}]

= - \frac{u^{6}}{24}

Substituting

u = \cos (x^{4})

, we get

- \frac{\cos^{6} (x^{4})}{24}

∴ \int x^{3} \sin (x^{4}) \cos^{5} (x^{4}) d x = - \frac{\cos^{6} (x^{4})}{24} + C

,
where C is the constant of integration.

Heather Fulton

Beginner2021-12-31Added 31 answers

\int x^{3} \sin (x^{4}) \cos^{5} (x^{4}) d x

4 \cdot x^{3} d x = d (x^{4}), t = x^{4}

\int \sin (t) \cdot \frac{{\cos (t)}^{5}}{4} d t

(- \sin (x)) d x = d (\cos (x)), u = \cos (x)

\int (- \frac{u^{5}}{4}) d u

\int - \frac{u^{5}}{4} d u = - \frac{u^{6}}{24} + C

- \frac{{\cos (t)}^{6}}{24} + C

- \frac{{\cos (x^{4})}^{6}}{24} + C

Vasquez

Expert2022-01-07Added 669 answers

$\begin{matrix} \int x^{3} * \sin (x^{4}) \cos^{5} (x^{4}) d x \\ T r a n s f o r m \\ \int - \frac{t^{5}}{4} d t \\ - \frac{1}{4} * \int t^{5} d t \\ - \frac{1}{4} * \frac{t^{6}}{6} \\ S u b s t i t u t e b a c k \\ - \frac{1}{4} * \frac{\cos (x^{4})^{6}}{6} \\ M u l t i p l y \\ - \frac{\cos (x^{4})^{6}}{24} \\ A d d C \\ S o l u t i o n \\ - \frac{\cos (x^{4})^{6}}{24} + C \end{matrix}$

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