What is the integral of sin(7x) ?

The integral of sin(7x) is -1/7 cos(7x)+C

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integral of sin(7x)

\int sin (7 x) d x

- \frac{1}{7} cos (7 x) + C

Solution steps

\int sin (7 x) d x

Apply u-substitution

: \int sin (u) \frac{1}{7} d u

= \int sin (u) \frac{1}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{7} (- cos (u))

Substitute back

u = 7 x

= \frac{1}{7} (- cos (7 x))

Simplify

= - \frac{1}{7} cos (7 x)

Add a constant to the solution

= - \frac{1}{7} cos (7 x) + C

y^{^{''}} + 4 y = 4 t

f (x) = (7^{- 3 x})

\int (4 x^{2} - \frac{2}{x}) d x

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

\int 1 + 4 x^{2} d x