Question

Find The Indefinite Integral. (Use C For The Constant Of Integration.) ∫Sin4xsin3xdx

Answer 1

`(1)/(2)\sin(x)-(1)/(14) \sin(7x)`

Explanation:

Evaluate the given integral.

`\Big\int\big(\sin(4x)\sin(3x)\big) \ dx`

`\hrulefill`

(1) - Apply the sum-to-product identity to the integrand

`\boxed{\left\begin{array}{ccc}\text{\underline{Sum-to-Product Identity:}}\\\\\sin(A)\sin(B)=(1)/(2)\Big(\cos(A-B)-\cos(A+B)\Big) \end{array}\right}`

`\Big\int\big(\sin(4x)\sin(3x)\big) \ dx\\\\\\\Longrightarrow \int\Big[(1)/(2)\Big(\cos(4x-3x)-\cos(4x+3x)\Big) \Big] \ dx\\\\\\\Longrightarrow \int\Big[(1)/(2)\Big(\cos(x)-\cos(7x)\Big) \Big] \ dx\\\\\\\Longrightarrow (1)/(2)\int\Big(\cos(x)-\cos(7x)\Big) \ dx`

(2) - We can now apply simple integration rules and use u-substitution

`\boxed{\left\begin{array}{ccc}\text{\underline{Trig. Int. Rule for Cosine:}}\\\\\int\cos(x) dx=\sin(x)\end{array}\right}`

`(1)/(2)\int\Big(\cos(x)-\cos(7x)\Big) \ dx \\\\ \\\text{Let} \ u=7x \rightarrow du=7dx \\ \\\\\Longrightarrow (1)/(2)\Big[\sin(x)-(1)/(7) \int\cos(u)du\Big]\\\\\\\Longrightarrow (1)/(2)\Big[\sin(x)-(1)/(7) \sin(7x)\Big]\\\\\\\Longrightarrow (1)/(2)\sin(x)-(1)/(14) \sin(7x)\Big\\\\\\\therefore \Big\int\big(\sin(4x)\sin(3x)\big) \ dx=\boxed{\boxed{(1)/(2)\sin(x)-(1)/(14) \sin(7x)}}`

Thus, the problem is solved.