Question

The value of ${\int }_a^{a+\left(\pi /2\right)}({\sin }^{4}x+{\cos }^{4}x)dx$ is

• A. independent of $a$
• B. $a(\frac{\pi }{2}{)}^2$
• C. $\frac{3\pi }{8}$
• D. $\frac{3\pi a^2}{8}$

Answer 1

Answer: (A), (C)

The correct options are
A independent of $a$
C $\frac{3\pi }{8}$

${\int }_a^{(a+\pi /2)}(si{n}^{4}x+co{s}^{4}x)dx$
${\int }_a^{a+\pi /2}(si{n}^{4}x+co{s}^{4}x+2co{s}^{2}xsi{n}^{2}x-2co{s}^{2}xsi{n}^{2}x)dx$
${\int }_a^{a+\pi /2}(si{n}^{2}x+co{s}^{2}x{)}^{2}dx-{\int }_a^{a+\pi /2}\frac{si{n}^{2}2x}{2}dx$
$=\frac{\pi }{2}-\frac{\pi }{8}=\frac{3\pi }{8}$, which is independent of $a$.
Hence, options 'A' and 'C' are correct.