Solve: \int \sin(y)\cos(y)dy

Question:

Solve:

{eq}\displaystyle \int \sin(y)\cos(y)\;dy {/eq}

Integration By Substitution:

To find the integral apply the integration by substitution. Consider, {eq}u = \cos(y) {/eq} while using the integration by substitution. Apply the power rule while integrating. The power rule is {eq}{\color{Blue} {\displaystyle \int y^n dx = \frac{y^{n + 1}}{n + 1}, where \ n \neq -1 }} {/eq}.

Answer and Explanation:


{eq}\displaystyle \int \sin (y)\cos (y) dx \\ \text{Apply u-substitution:} \\ u = \sin(y) \\ du = \cos(y) dx \\ \begin{align*} \int \sin (y)\cos (y) dy &= \int u du \\ &= \frac{u^2}{2} &{\color{Blue}{\text{(Apply the power rule:} \displaystyle \int y^n dx = \frac{y^{n + 1}}{n + 1}, where \ n \neq -1)}} \\ &= \frac{\sin^2(y)}{2} &\text{(Where,} \ u = \sin(y)) \\ &= \frac{1}{2}\sin^2(y) + C &\text{(Add the constant)} \end{align*} {/eq}


{eq}\text{Therefore, the solution is} \displaystyle {\color{Red}{\int \sin (y)\cos (y) dy = \frac{1}{2}\sin^2(y) + C}} {/eq}


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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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