Definite Integrals

Question

Let $f(x)$ be a continuous function satisfying $f(x)f(a-x) = 1$. Then $\int_0^a \frac{1}{1 + f(x)} dx$ equals:

RankGuru · Accepted Answer

Let $I = \int_0^a \frac{1}{1 + f(x)} dx$.
Using the property $\int_0^a g(x) dx = \int_0^a g(a-x) dx$:
$I = \int_0^a \frac{1}{1 + f(a-x)} dx$.

Given $f(x)f(a-x) = 1$, we imply $f(a-x) = 1/f(x)$.
Substitute this back:
$I = \int_0^a \frac{1}{1 + 1/f(x)} dx = \int_0^a \frac{f(x)}{f(x) + 1} dx$.

Now add the original $I$ and the new form:
$2I = \int_0^a \frac{1}{1 + f(x)} dx + \int_0^a \frac{f(x)}{1 + f(x)} dx$
$2I = \int_0^a \frac{1 + f(x)}{1 + f(x)} dx = \int_0^a 1 dx = a$.
Thus, $I = a/2$.

Rank Guru

Definite Integrals

Select the correct option:

Solution

🔒 Solution Hidden from View

More definite integrals Practice Questions

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:

The value of $lim_{n \to \infty} \sum_{r = 1}^{n} \frac{1}{4 n ^{2} - r ^{2}}$ is:

Evaluate $\int e^{x} (\frac{1 + x l n x}{x}) d x$ (Assume $x > 0$ and base $e$ for $ln$ ).

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:

About This Question

Definite Integrals

Select the correct option:

Solution

🔒 Solution Hidden from View

More definite integrals Practice Questions

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:

The value of limn→∞​∑r=1n​4n2−r2​1​ is:

Evaluate ∫ex(x1+xlnx​)dx (Assume x>0 and base e for ln).

The value of ∫0π/2​sin2025x+cos2025xsin2025x​dx is:

About This Question

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:

The value of $lim_{n \to \infty} \sum_{r = 1}^{n} \frac{1}{4 n ^{2} - r ^{2}}$ is:

Evaluate $\int e^{x} (\frac{1 + x l n x}{x}) d x$ (Assume $x > 0$ and base $e$ for $ln$ ).

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: