Question

Evaluate ${\int }_1^4\left(cos\right(x)-\frac{3}{x^5})dx$

• A. $\sin 4-\sin 1-\frac{765}{1024}$
• B. $\sin 4-\sin 1-\frac{756}{1024}$
• C. $\sin 4-\sin 1-\frac{675}{1024}$
• D. $2\sin 4-\sin 1-\frac{765}{1024}$

Answer 1

The correct option is A $\sin 4-\sin 1-\frac{765}{1024}$

We can easily observe that at $x=0$ second part of integration $\left(\frac{3}{x^5}\right)$ is undefined but it doesn’t lie in the interval of 1 to 4. Hence, we can directly integrate the whole function without breaking it.
${\int }_0^4\left(cos\right(x)-\frac{3}{x^5})dx={\int }_0^4\left(cos\right(x)-3x^{-5})dx$
$[\sin x{]}_1^4+{\left[\frac{3}{4x^4}\right]}_1^4=\sin 4-\sin 1-\frac{765}{1024}$