What is the integral of 10sin(5x) ?

The integral of 10sin(5x) is -2cos(5x)+C

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integral of 10sin(5x)

\int 10 sin (5 x) d x

- 2 cos (5 x) + C

Solution steps

\int 10 sin (5 x) d x

Take the constant out:

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

= 10 \cdot \int sin (5 x) d x

Apply u-substitution

: \int sin (u) \frac{1}{5} d u

= 10 \cdot \int sin (u) \frac{1}{5} d u

Take the constant out:

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

= 10 \cdot \frac{1}{5} \cdot \int sin (u) d u

Use the common integral:

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

= 10 \cdot \frac{1}{5} (- cos (u))

Substitute back

u = 5 x

= 10 \cdot \frac{1}{5} (- cos (5 x))

Simplify

10 \cdot \frac{1}{5} (- cos (5 x)) : - 2 cos (5 x)

= - 2 cos (5 x)

Add a constant to the solution

= - 2 cos (5 x) + C

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

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

\frac{2 x}{y} d x - \frac{x ^{2}}{y ^{2}} d y = 0

f (x) = \frac{2 x}{7} + 1