Question

Cálculo I - Integral. ∫sen(5x)cos(5x) dx

Answer 1

Explicação passo-a-passo:

Cálculo da integral indefinida

Dada a integral :

$\red{ \displaystyle\int \sf{ \sin(5x) * \cos(5x) dx} }$

Como: $\sf{ \blue{ \sin(\alpha)*\cos(b)~=~ \dfrac{1}{2}*\left( \sin(\alpha + b) + \sin( \alpha - b) \right) } }$

Então :

$\iff \sf{ I~=~ } \displaystyle\int \sf{ \dfrac{1}{2}\left( \sin(5x + 5x) + \sin( 5x - 5x) \right)dx}$

$\iff \sf{ I~=~ \dfrac{1}{2}}\displaystyle\int \sf{ \sin(10x) + \sin(0)dx }$

$\iff \sf{ I~=~ -\dfrac{1}{2} *\dfrac{1}{10}}\displaystyle\int \sf{ -10\sin(10x) dx }$

$\green{ \iff \boxed{ \boxed{ \sf{ I~=~ -\dfrac{\cos(10x)}{20}+c~,com~c\in \mathbb{R} } } } \sf{ \longleftarrow Resposta } }$

Alternativa A)

Espero ter ajudado bastante!)