Definite Integrals

Question

Let $f: \mathbb{R} 	o \mathbb{R}$ be a differentiable function such that $f(x) = x^2 + \int_0^x e^{-(x-t)} f(t) dt$. Then $f(x)$ equals:

RankGuru · Accepted Answer

Note: The integral term can be simplified by factorization logic using $e^{-(x-t)} = e^{-x}e^t$. We use differentiation.

Method: Differentiate both sides.
given $f(x) = x^2 + \int_0^x e^{-(x-t)} f(t) dt$
$f(x) = x^2 + e^{-x} \int_0^x e^t f(t) dt$.
Multiply by $e^x$:
$e^x f(x) = x^2 e^x + \int_0^x e^t f(t) dt$.
Differentiate wrt $x$:
$e^x f'(x) + e^x f(x) = (x^2 e^x + 2x e^x) + e^x f(x)$.
Cancel $e^x f(x)$:
$e^x f'(x) = e^x (x^2 + 2x)$.
divide by $e^x$:
$f'(x) = x^2 + 2x$.
Integrate:
$f(x) = x^3/3 + x^2 + C$.
From original eq, $f(0) = 0$. So $C=0$.
$f(x) = x^3/3 + x^2$.

Rank Guru

Definite Integrals

Select the correct option:

Solution

🔒 Solution Hidden from View

More definite integrals Practice Questions

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 $ln$ ).

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

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

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).

Let f(x) be a continuous function satisfying f(x)f(a−x)=1. Then $\int_0^a \frac{1}{1 + f(x)} d...

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

About This Question

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 $ln$ ).

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

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: