What is the integral of 1/2 tcos(t^2) ?

The integral of 1/2 tcos(t^2) is 1/4 sin(t^2)+C

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integral of 1/2 tcos(t^2)

\int \frac{1}{2} t cos (t^{2}) d t

\frac{1}{4} sin (t^{2}) + C

Solution steps

\int \frac{1}{2} t cos (t^{2}) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int t cos (t^{2}) d t

Apply u-substitution

: \int \frac{cos ( u )}{2} d u

= \frac{1}{2} \cdot \int \frac{cos ( u )}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{2} \cdot \frac{1}{2} sin (u)

Substitute back

u = t^{2}

= \frac{1}{2} \cdot \frac{1}{2} sin (t^{2})

Simplify

\frac{1}{2} \cdot \frac{1}{2} sin (t^{2}) : \frac{1}{4} sin (t^{2})

= \frac{1}{4} sin (t^{2})

Add a constant to the solution

= \frac{1}{4} sin (t^{2}) + C

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What is the integral of 1/2 tcos(t^2) ?
The integral of 1/2 tcos(t^2) is 1/4 sin(t^2)+C