What is the integral of 5xcos(4x^2+2) ?

The integral of 5xcos(4x^2+2) is 5/8 sin(4x^2+2)+C

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integral of 5xcos(4x^2+2)

\int 5 x cos (4 x^{2} + 2) d x

\frac{5}{8} sin (4 x^{2} + 2) + C

Solution steps

\int 5 x cos (4 x^{2} + 2) d x

Take the constant out:

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

= 5 \cdot \int x cos (4 x^{2} + 2) d x

Apply u-substitution

= 5 \cdot \int \frac{cos ( u )}{8} d u

Take the constant out:

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

= 5 \cdot \frac{1}{8} \cdot \int cos (u) d u

Use the common integral:

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

= 5 \cdot \frac{1}{8} sin (u)

Substitute back

u = 4 x^{2} + 2

= 5 \cdot \frac{1}{8} sin (4 x^{2} + 2)

Simplify

5 \cdot \frac{1}{8} sin (4 x^{2} + 2) : \frac{5}{8} sin (4 x^{2} + 2)

= \frac{5}{8} sin (4 x^{2} + 2)

Add a constant to the solution

= \frac{5}{8} sin (4 x^{2} + 2) + C

What is the integral of 5xcos(4x^2+2) ?
The integral of 5xcos(4x^2+2) is 5/8 sin(4x^2+2)+C