What is the integral of 7x^3cos(x^2) ?

The integral of 7x^3cos(x^2) is 7/2 (x^2sin(x^2)+cos(x^2))+C

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integral of 7x^3cos(x^2)

\int 7 x^{3} cos (x^{2}) d x

\frac{7}{2} (x^{2} sin (x^{2}) + cos (x^{2})) + C

Solution steps

\int 7 x^{3} cos (x^{2}) d x

Take the constant out:

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

= 7 \cdot \int x^{3} cos (x^{2}) d x

Apply u-substitution

= 7 \cdot \int \frac{u cos ( u )}{2} d u

Take the constant out:

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

= 7 \cdot \frac{1}{2} \cdot \int u cos (u) d u

Apply Integration By Parts

= 7 \cdot \frac{1}{2} (u sin (u) - \int sin (u) d u)

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

= 7 \cdot \frac{1}{2} (u sin (u) - (- cos (u)))

Substitute back

u = x^{2}

= 7 \cdot \frac{1}{2} (x^{2} sin (x^{2}) - (- cos (x^{2})))

Simplify

7 \cdot \frac{1}{2} (x^{2} sin (x^{2}) - (- cos (x^{2}))) : \frac{7}{2} (x^{2} sin (x^{2}) + cos (x^{2}))

= \frac{7}{2} (x^{2} sin (x^{2}) + cos (x^{2}))

Add a constant to the solution

= \frac{7}{2} (x^{2} sin (x^{2}) + cos (x^{2})) + C

What is the integral of 7x^3cos(x^2) ?
The integral of 7x^3cos(x^2) is 7/2 (x^2sin(x^2)+cos(x^2))+C