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\int s^{\frac{1}{3}} \cos \left(s^{\frac{4}{3}}-8\right) d s

Feb 1, 2024

\int s^{\frac{1}{3}} \cos \left(s^{\frac{4}{3}}-8\right) d s

Solution by Steps

step 1

To integrate the function

s^{\frac{1}{3}} \cos \left(s^{\frac{4}{3}}-8\right)

with respect to

s

, we look for a substitution that simplifies the integral

step 2

Let

u = s^{\frac{4}{3}}

. Then,

\frac{du}{ds} = \frac{4}{3} s^{\frac{1}{3}}

, which implies

ds = \frac{3}{4} s^{-\frac{1}{3}} du

step 3

Substituting

u

and

ds

into the integral, we get

\int s^{\frac{1}{3}} \cos(u - 8) \cdot \frac{3}{4} s^{-\frac{1}{3}} du

, which simplifies to

\frac{3}{4} \int \cos(u - 8) du

step 4

The integral of

\cos(u - 8)

with respect to

u

\sin(u - 8)

step 5

Multiplying by the constant

\frac{3}{4}

, we get

\frac{3}{4} \sin(u - 8) + C

, where

C

is the constant of integration

step 6

Substituting back for

u

, we have

\frac{3}{4} \sin(s^{\frac{4}{3}} - 8) + C

Answer

-\frac{3}{4} \sin(8 - s^{\frac{4}{3}}) + C

Key Concept

Integration by Substitution

Explanation

The integral was solved by making a substitution that simplifies the integral, performing the integration, and then substituting back to the original variable.