What is the integral of 2xcos(x^2-5) ?

The integral of 2xcos(x^2-5) is sin(x^2-5)+C

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integral of 2xcos(x^2-5)

\int 2 x cos (x^{2} - 5) d x

sin (x^{2} - 5) + C

Solution steps

\int 2 x cos (x^{2} - 5) d x

Take the constant out:

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

= 2 \cdot \int x cos (x^{2} - 5) d x

Apply u-substitution

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

= 2 \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

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

Use the common integral:

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

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

Substitute back

u = x^{2} - 5

= 2 \cdot \frac{1}{2} sin (x^{2} - 5)

Simplify

2 \cdot \frac{1}{2} sin (x^{2} - 5) : sin (x^{2} - 5)

= sin (x^{2} - 5)

Add a constant to the solution

= sin (x^{2} - 5) + C

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What is the integral of 2xcos(x^2-5) ?
The integral of 2xcos(x^2-5) is sin(x^2-5)+C