Middle Tennessee State University Integration Question

Question Description

Help me study for my Calculus class. I’m stuck and don’t understand.

Two calculus students were asked to evaluate LaTeX: \displaystyle \int \sin x \cos x \; dx.

∫sin⁡xcos⁡xdx.

Student A arrived at an answer of LaTeX: \displaystyle \frac {\sin^2 x}{2} + C

sin2⁡x2+C

, while Student B arrived at an answer of LaTeX: \displaystyle -\frac {\cos^2 x}{2} + C.\;

−cos2⁡x2+C.

Both students received full credit for their responses.

Given that LaTeX: \displaystyle \frac {\sin^2 x}{2} \neq -\frac {\cos^2 x}{2}\;

sin2⁡x2≠−cos2⁡x2

it appears that the students arrived at completely different answers.

Explain why either representation for LaTeX: \displaystyle \int \sin x \cos x \; dx\;

∫sin⁡xcos⁡xdx

is valid. You may want to consider the graphs of LaTeX: \displaystyle y = \frac {\sin^2 x}{2}\;

y=sin2⁡x2

and LaTeX: \displaystyle -\frac {\cos^2 x}{2}\;

−cos2⁡x2

in your analysis. Pleas use word file to answer the question.