Question

resolva a integral ∫ 4x^3 sen(3x^4-5) dx

Answer 1

$\int4x^{3}sen(3x^{4}-5)dx$

Seja u = 3x⁴ - 5:

$u=3x^{4}-5~~~~~~~~~(derivando:)\\du=3\cdot4x^{3}dx~~~~~~(isolando~4x^{3}dx:)\\\\\\\boxed{\boxed{4x^{3}dx=\dfrac{1}{3}du}}$

Então:

$\int4x^{3}sen(3x^{4}-5)dx=\int sen(3x^{4}-5)\cdot4x^{3}dx\\\\\int4x^{3}sen(3x^{4}-5)dx=\int sen(u)\cdot\frac{1}{3}du\\\\\int4x^{3}sen(3x^{4}-5)dx=\frac{1}{3}\int sen(u)du\\\\\int4x^{3}sen(3x^{4}-5)dx=\frac{1}{3}(-cos~(u))+C$

Voltando para a variável x:

$\boxed{\boxed{\int4x^{3}sen(3x^{4}-5)dx=-\dfrac{cos(3x^{4}-5)}{3}+C}}$