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

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

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integral of 5xcos(x^2)

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

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

Solution steps

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

Apply u-substitution

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

= 5 \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

= 5 \cdot \frac{1}{2} \cdot \int cos (u) d u

Use the common integral:

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

= 5 \cdot \frac{1}{2} sin (u)

Substitute back

u = x^{2}

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

Simplify

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

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

Add a constant to the solution

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

\frac{d y}{d x} = \frac{1}{x} (y^{2} + 1 y)

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

\int sin (x) cos (x) d x

\int_{0}^{5} \frac{1}{x ^{\frac{14}{13}}} d x

f (x) = 3 x^{5} e^{2 x}

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