Find the value of _0^π/2 ^8 x/^8 x + ^8 x dx . - Mathematics MCQ Mathematics Mock Test - 1 2026 | ExamTest.live

mediumMCQMathematics Mock Test - 12026Mathematics

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Find the value of \int_{0}^{π /2} \frac{s i n ^{8} x}{s i n ^{8} x + c o s ^{8} x} d x .

Solution & Step-by-step Explanation

Let I = \int_{0}^{π /2} \frac{s i n ^{8} x}{s i n ^{8} x + c o s ^{8} x} d x —(1) Using the property \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x : I = \int_{0}^{π /2} \frac{sin ^{8} ( π /2 - x )}{sin ^{8} ( π /2 - x ) + cos ^{8} ( π /2 - x )} d x I = \int_{0}^{π /2} \frac{cos ^{8} x}{cos ^{8} x + sin ^{8} x} d x —(2) Adding (1) and (2): 2 I = \int_{0}^{π /2} \frac{sin ^{8} x + cos ^{8} x}{sin ^{8} x + cos ^{8} x} d x 2 I = \int_{0}^{π /2} 1 \cdot d x 2 I = [x]_{0}^{π /2} = \frac{π}{2} I = \frac{π}{4}

Try it yourself before checking the explanation above.

Find the value of \int_{0}^{π /2} \frac{s i n ^{8} x}{s i n ^{8} x + c o s ^{8} x} d x .

\frac{π}{4}

\frac{π}{2}

0

\frac{3 π}{4}

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