What is the integral of x*e^{sin(x^2)}cos(x^2) ?

The integral of x*e^{sin(x^2)}cos(x^2) is 1/2 e^{sin(x^2)}+C

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integral of x*e^{sin(x^2)}cos(x^2)

\int x \cdot e^{s i n (x^{2})} cos (x^{2}) d x

\frac{1}{2} e^{s i n (x^{2})} + C

Solution steps

\int x e^{s i n (x^{2})} cos (x^{2}) d x

Apply u-substitution

: \int \frac{e ^{s i n (u)} cos ( u )}{2} d u

= \int \frac{e ^{s i n (u)} 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 \int e^{s i n (u)} cos (u) d u

Apply u-substitution

: \int e^{v} d v

= \frac{1}{2} \cdot \int e^{v} d v

Use the common integral:

\int e^{v} d v = e^{v}

= \frac{1}{2} e^{v}

Substitute back

= \frac{1}{2} e^{s i n (x^{2})}

Add a constant to the solution

= \frac{1}{2} e^{s i n (x^{2})} + C

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What is the integral of x*e^{sin(x^2)}cos(x^2) ?
The integral of x*e^{sin(x^2)}cos(x^2) is 1/2 e^{sin(x^2)}+C