Can u integrate it

Calculus Level 3

If $\displaystyle f(x) = \int \cos (2018x) \sin^{2016} x \ dx$ , $f(0) = 0$ and $f \left(\dfrac \pi 4\right) = \dfrac \alpha{\beta \gamma^\delta}$ , where $\alpha$ , $\beta$ , $\gamma$ and $\delta$ are positive integers with $\beta$ , $\gamma$ and $\delta$ being primes. Find $\alpha +\beta +\gamma +\delta$ .

The answer is 3029.

1 solution

Chew-Seong Cheong
Mar 1, 2018

$\begin{aligned} f(x) & = \int \cos (2018x) \sin^{2016} x \ dx \\ & = \int \cos (2017x+x) \sin^{2016} x \ dx \\ & = \int \left(\cos (2017x)\cos x\sin^{2016}x - \sin (2017x) \sin^{2017} x\right) \ dx \\ & = \frac 1{2017} \int \frac d{dx} \cos (2017x)\sin^{2017} x \ dx \\ & = \frac 1{2017} \int d \cos (2017x)\sin^{2017} x \\ & = \frac {\cos (2017x)\sin^{2017} x}{2017} + \color{#3D99F6} C & \small \color{#3D99F6} \text{where }C \text{ is the constant of integration.} \\ f(0) & = 0 + C = 0 & \small \color{#3D99F6} \implies C = 0 \end{aligned}$

$\begin{aligned} \implies f\left(\frac \pi 4\right) & = \frac {\cos \left(2017\times \frac \pi 4\right)\sin^{2017} \frac \pi 4}{2017} = \frac {\cos \left(\frac \pi 4\right)\sin^{2017} \frac \pi 4}{2017} = \frac {\left(\frac 1{\sqrt 2}\right)^{2018}}{2017} = \frac 1{2017\times 2^{1009}} \end{aligned}$

Therefore, $\alpha + \beta + \gamma + \delta = 1+2017+2+1009 = \boxed{3029}$ .

@Ayush Mishra , just like other functions such as \frac, \int you need to add a backslash before \sin, \cos, \tan, \sec, \cot, \csc. Note that \sin x $\sin x$ , the sin is not in italic as a function should be and there is a space automatically inserted between the operator $\sin$ and the operand $x$ (in italic as a variable or constant) without the backslash sin x $sin x$ , note that all sin x are in italic and there is no space between them. You don't need to add \quad. The creators of LaTex are of course smart enough as not to make the coding difficult.

Chew-Seong Cheong - 3 years, 3 months ago

thank u sir I will keep it in mind from next time

A Former Brilliant Member - 3 years, 3 months ago

You can also use this editor to learn the codes. Daum Equation Editor

Don't use it to code everything. That way you won't learn.

Chew-Seong Cheong - 3 years, 3 months ago

Can u integrate it

The answer is 3029.

1 solution

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