Question

find ∫sin(2πx + 30°)dx​

Answer 1

GIVEN :–

• A Function sin(2πx + 30°).

TO FIND :–

• Integration with respect to 'x' of given function = ?

SOLUTION :–

• Let –

$\\ \implies \bf \mathbb{A} \: = \int \sin(2\pi x + {30}^{ \circ} ).dx$

• Let's put –

$\\ \longrightarrow \: \: \bf 2\pi x + {30}^{ \circ} = t$

$\\ \longrightarrow \: \: \bf 2\pi dx =dt$

$\\ \longrightarrow \: \: \bf dx = \dfrac{dt}{2\pi}$

$\\ \implies \bf \mathbb{A} \: = \int \sin(t) \dfrac{dt}{2\pi}$

$\\ \implies \bf \mathbb{A} \: = \dfrac{1}{2\pi} \int \sin(t) dt$

• We know that –

$\\ \: \: \bigstar \: \: \: \: \pink{\boxed{ \bf\int \sin(t) dt = - \cos(x) + c }}$

• So that –

$\\ \implies \bf \mathbb{A} \: = - \dfrac{1}{2\pi} \cos(t) + c$

• Now , Replace 't' –

$\\ \implies \green{\large { \boxed{\bf \mathbb{A} \: = - \dfrac{1}{2\pi} \cos(2\pi x + {30}^{\circ} ) + c}}}$

• Hence –

$\\ \implies \red{\large { \boxed{\bf \int \sin(2\pi x + {30}^{ \circ} ).dx \: = - \dfrac{1}{2\pi} \cos(2\pi x + {30}^{\circ} ) + c}}}$