What is the integral of 2xsin(x^2+1) ?

The integral of 2xsin(x^2+1) is -cos(x^2+1)+C

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integral of 2xsin(x^2+1)

\int 2 x sin (x^{2} + 1) d x

- cos (x^{2} + 1) + C

Solution steps

\int 2 x sin (x^{2} + 1) d x

Take the constant out:

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

= 2 \cdot \int x sin (x^{2} + 1) d x

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

u = x^{2} + 1

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

Simplify

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

= - cos (x^{2} + 1)

Add a constant to the solution

= - cos (x^{2} + 1) + C

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What is the integral of 2xsin(x^2+1) ?
The integral of 2xsin(x^2+1) is -cos(x^2+1)+C