What is the integral of 3x*cos(3-x^2) ?

The integral of 3x*cos(3-x^2) is 3/2 sin(-3+x^2)+C

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integral of 3x*cos(3-x^2)

\int 3 x \cdot cos (3 - x^{2}) d x

\frac{3}{2} sin (- 3 + x^{2}) + C

Solution steps

\int 3 x cos (3 - x^{2}) d x

Take the constant out:

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

= 3 \cdot \int x cos (3 - x^{2}) d x

Factor

3 - x^{2} : - (- 3 + x^{2})

= 3 \cdot \int x cos (- (- 3 + x^{2})) d x

Simplify

x cos (- (- 3 + x^{2})) : x cos (- 3 + x^{2})

= 3 \cdot \int x cos (- 3 + x^{2}) d x

Apply u-substitution

= 3 \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

= 3 \cdot \frac{1}{2} \cdot \int cos (u) d u

Use the common integral:

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

= 3 \cdot \frac{1}{2} sin (u)

Substitute back

u = - 3 + x^{2}

= 3 \cdot \frac{1}{2} sin (- 3 + x^{2})

Simplify

3 \cdot \frac{1}{2} sin (- 3 + x^{2}) : \frac{3}{2} sin (- 3 + x^{2})

= \frac{3}{2} sin (- 3 + x^{2})

Add a constant to the solution

= \frac{3}{2} sin (- 3 + x^{2}) + C

What is the integral of 3x*cos(3-x^2) ?
The integral of 3x*cos(3-x^2) is 3/2 sin(-3+x^2)+C