Question

The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\)(x³+xcosx+tan⁵x+1)dx is

• A. 0
• B. 2
• C. \(\pi\)
• D. 1

Answer 1

The Correct Option is C

Let I=\(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\)(x³+xcosx+tan⁵x+1)dx

⇒I=\(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\)x³+dx+\(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\)cosx+\(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\)tan⁵xdx+\(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\)1. dx

It is known that if f(x)is an even function,then\(\int_{-a}^{a}\)ƒ(x)dx=2\(\int_{0}^{a}\)ƒ(x)dx and

if f(x)is an odd function,then∫a-aƒ(x)dx=0

I=0+0+0+2\(\int_{0}^{\frac{\pi}{2}}\)1.dx

=2[x]\(^{\frac{\pi}{2}}_{2}\)

=\(\frac{2\pi}{2}\)

=\(\pi\)

Hence,the correct Answer is C.