The value of $\int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx$ is:

Let $I = \int_0^{\pi/2} \frac{\sin^{n} x}{\sin^{n} x + \cos^{n} x} dx$. Using Property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$: Replace $x$ with $\pi/2 - x$. Note that $\sin(\pi/2 - x) = \cos x$ and $\cos(\pi/2 - x) = \sin x$. $I = \int_0^{\pi/2} \frac{\cos^{n} x}{\cos^{n} x + \sin^{n} x} dx$. Adding the two expressions for $I$: $2I = \int_0^{\pi/2} \frac{\sin^n x + \cos^n x}{\sin^n x + \cos^n x} dx$ $2I = \int_0^{\pi/2} 1 dx = [x]_0^{\pi/2} = \pi/2$. So, $I = \pi/4$.

Definite Integrals

Easymathematics

The value of $\int_{0}^{π /2} \frac{s i n ^{2025} x}{s i n ^{2025} x + c o s ^{2025} x} d x$ is:

Select the correct option:

$π$

$π /2$

$π /4$

$0$

🔒 Solution Hidden from View

Submit your answer to unlock the detailed step-by-step solution.

About This Question

Subject: mathematics
Chapter: integral calculus
Topic: definite integrals
Difficulty: Easy
Year: 2025

Solution

Correct Answer:

$π /4$

Let $I = \int_{0}^{π /2} \frac{s i n ^{n} x}{s i n ^{n} x + c o s ^{n} x} d x$ . Using Property $\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$ : Replace $x$ with $π /2 - x$ . Note that $sin (π /2 - x) = cos x$ and $cos (π /2 - x) = sin x$ . $I = \int_{0}^{π /2} \frac{c o s ^{n} x}{c o s ^{n} x + s i n ^{n} x} d x$ .

Adding the two expressions for $I$ : $2 I = \int_{0}^{π /2} \frac{s i n ^{n} x + c o s ^{n} x}{s i n ^{n} x + c o s ^{n} x} d x$ $2 I = \int_{0}^{π /2} 1 d x = [x]_{0}^{π /2} = π /2$ . So, $I = π /4$ .

This easy difficulty mathematics question is from the chapter integral calculus, covering the topic of definite integrals. It appeared in the 2025 exam.

Looking for more practice? Explore all mathematics questions or browse integral calculus questions on RankGuru.

Definite Integrals

Select the correct option:

Solution

🔒 Solution Hidden from View

More definite integrals Practice Questions

Assertion: the integral from zero to two of the function defined as x for x below one and as two min...

Evaluate the definite integral of \sin^2 x from 0 to \pi/2 and identify which standard symmetry resu...

Find the indefinite integral of the function 1 divided by the product x times the natural logarithm ...

Using the king property of definite integrals, evaluate the integral from 0 to \pi of x times \sin x...

Let $f : R \to R$ be a differentiable function such that $f(x) = x^2 + \int_0^x e^{...

If $I_{n} = \int_{0}^{π /4} tan^{n} x d x$ (for $n > 1$ ), then $I_{n} + I_{n - 2}$ equals:

About This Question

Solution

Definite Integrals

Select the correct option:

Solution

🔒 Solution Hidden from View

More definite integrals Practice Questions

Assertion: the integral from zero to two of the function defined as x for x below one and as two min...

Evaluate the definite integral of \sin^2 x from 0 to \pi/2 and identify which standard symmetry resu...

Find the indefinite integral of the function 1 divided by the product x times the natural logarithm ...

Using the king property of definite integrals, evaluate the integral from 0 to \pi of x times \sin x...

Let f:R→R be a differentiable function such that $f(x) = x^2 + \int_0^x e^{...

If In​=∫0π/4​tannxdx (for n>1), then In​+In−2​ equals:

About This Question

Solution

Let $f : R \to R$ be a differentiable function such that $f(x) = x^2 + \int_0^x e^{...

If $I_{n} = \int_{0}^{π /4} tan^{n} x d x$ (for $n > 1$ ), then $I_{n} + I_{n - 2}$ equals: