1. Tardigrade
  2. Question
  3. Mathematics
  4. For f( x )= x 4+| x |, let I 1=∫ limits0π f( cos x ) dx and I 2=∫ limits0π / 2 f( sin x ) dx then ( I 1/ I 2) has the value equal to

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Q. For $f( x )= x ^4+| x |$, let $I _1=\int\limits_0^\pi f(\cos x ) dx$ and $I _2=\int\limits_0^{\pi / 2} f(\sin x ) dx$ then $\frac{ I _1}{ I _2}$ has the value equal to

Integrals

Solution:

Clearly f is an even function, hence
$I_1=\int\limits_0^\pi f\left(\cos (\pi-x) d x=\int\limits_0^\pi f(-\cos x) d x=\int_0^\pi f(\cos x) d x\right. $
$\therefore I_1=2 \int\limits_0^{\pi / 2} f(\cos x) d x=2 \int\limits_0^{\pi / 2} f(\sin x) d x=2 I_2 \Rightarrow \frac{I_1}{I_2}=2$