How can I solve the integral 2 ∫ t cos(t) dt using integration by parts?

[vande060]

Homework Statement

I have to solve this integral

S cos(x^1/2)dx

where S is the integral symbol

Homework Equations

The Attempt at a Solution

the book tells me to use substitution and then integrate by parts

so i say u = x^1/2
du = 1/2*x^-1/2

then i can write 2 S (cos(u)du)/ x^1/2

where S in the integral sign

from here i think i can substitute the x^1/2 in the denominator by u because of the definition u = x^1/2

after the last substitution my integral would look like 2 S cos(u)/u

is this even close to right

 

[dextercioby]

Homework Statement

I have to solve this integral

S cos(x^1/2)dx

where S is the integral symbol

Homework Equations

The Attempt at a Solution

the book tells me to use substitution and then integrate by parts

so i say u = x^1/2
du = 1/2*x^-1/2 dx

then i can write 2 S (cos(u)du)/ x^1/2

where S in the integral sign

from here i think i can substitute the x^1/2 in the denominator by u because of the definition u = x^1/2

after the last substitution my integral would look like 2 S cos(u)/u du

is this even close to right

There. Fixed the missing part. You'll never get an elementary function instead of the question mark below

$$\int\frac{\cos u}{u} {}du= ? + C$$

However, the computation you made is wrong. <u> should be in the numerator, so the <exotic> part won't apply.

 

[AfterSunShine]

$$t=\sqrt{x} \implies t^2=x \implies 2tdt=dx$$

Your integral will be :

$$2 \int \, t \, cos(t) \, dt$$

A quick application of integration by parts will kill it.

 

[vande060]

$$t=\sqrt{x} \implies t^2=x \implies 2tdt=dx$$

Your integral will be :

$$2 \int \, t \, cos(t) \, dt$$

A quick application of integration by parts will kill it.

oh wow i didnt even think of that, good suggestion and thank you. i can finish the integration by parts no problem