What is the integral of sin(sqrt(5x+1)) ?

The integral of sin(sqrt(5x+1)) is 2/5 (-cos(sqrt(5x+1))sqrt(5x+1)+sin(sqrt(5x+1)))+C

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integral of sin(sqrt(5x+1))

\int sin (5 x + 1) d x

\frac{2}{5} (- cos (5 x + 1) 5 x + 1 + sin (5 x + 1)) + C

Solution steps

\int sin (5 x + 1) d x

Apply u-substitution

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

= \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

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

Apply u-substitution

: \int sin (v) \cdot 2 v d v

= \frac{1}{5} \cdot \int sin (v) \cdot 2 v d v

Take the constant out:

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

= \frac{1}{5} \cdot 2 \cdot \int sin (v) v d v

Apply Integration By Parts

: - v cos (v) - \int - cos (v) d v

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

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

= \frac{1}{5} \cdot 2 (- v cos (v) - (- sin (v)))

Substitute back

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

Simplify

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

= \frac{2}{5} (- cos (5 x + 1) 5 x + 1 + sin (5 x + 1))

Add a constant to the solution

= \frac{2}{5} (- cos (5 x + 1) 5 x + 1 + sin (5 x + 1)) + C

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What is the integral of sin(sqrt(5x+1)) ?
The integral of sin(sqrt(5x+1)) is 2/5 (-cos(sqrt(5x+1))sqrt(5x+1)+sin(sqrt(5x+1)))+C