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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 2x^3sin(x^2)

\int 2 x^{3} sin (x^{2}) d x

- x^{2} cos (x^{2}) + sin (x^{2}) + C

Solution steps

\int 2 x^{3} sin (x^{2}) d x

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

: - u cos (u) - \int - cos (u) d u

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

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

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

Substitute back

u = x^{2}

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

Simplify

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

= - x^{2} cos (x^{2}) + sin (x^{2})

Add a constant to the solution

= - x^{2} cos (x^{2}) + sin (x^{2}) + C

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

\int \frac{6 x ^{4} + 2 x ^{2} + 4 x + 15}{x ^{2} + 3} d x

\int \frac{2 t ^{2} - t + 4}{t ^{3} + 4 t} d t

\int 6 tan^{2} (x) sec^{4} (x) d x

\frac{1}{( x ^{2} + x x )}

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