$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Since $X$ and $Y$ are independent, and because $\mu=0$ and $\sigma=1$, then we know that $$f_{X,Y}(x,y)=f_X(x)f_Y(y)=\frac{1}{2\pi}\exp(\frac{-(x^2+y^2}{2}).$$
Of course, since our domain over which we are integrating is a circle with a radius of 1, we convert $x^2+y^2=r^2$, and $\text{d}x \text{d}y=r\text{d}r\text{d}\theta$, using $x=r\cos\theta$ and $y=r\sin \theta$ to get the standard polar coordinate transformation. Notice, we are converting the joint pdf $f_{X,Y}(x,y)$ to make our lives easier, and the circle defined by the question, ie $X^2+Y^2<1$, merely comes into play when looking at what we integrate over.

\begin{align*} P(X^2+Y^2<1)&=\frac{1}{2\pi}\int_{r=0}^{1}\int_{\theta=0}^{2\pi}e^{(\frac{-r^2}{2})}r \text{d}r \text{d}\theta\\ &=2\pi \frac{1}{2\pi}\int_{r=0}^{1}e^{(\frac{-r^2}{2})}r \text{d}r\\ &=\int_{u=1}^{e^{-1/2}}-\text{d}\underbrace{(e^{-r^2/2})}_u=-u\bigg|_1^{e^{-1/2}}\\ &=-e^{1/2}--1=1-e^{-1/2} \end{align*}

The u-substitution may add some confusion, especially with changing the limits. For this question, it may be more obvious to just skip the explicit substitution, and just leave $-\text{d}e^{-r^2/2}$ with $e^{-r^2/2}$ the variable in the differential term, as that may be clearer and more obvious, as this is a well known derivative.

Although the following method is more convoluted than necessary, I think it provides a fun way to see the convolution method for computing the density of a sum of independent random variables in action. It also requires a few fun integration tricks.

First we will find the distribution for $X^2$ (and, by symmetry, the distribution of $Y^2$). We consider the cdf $F_{X^2}(t)$, and we see that $$F_{X^2}(t) = P(X^2 \leq t) = P(|X| \leq \sqrt{t}) = P(X \in [-\sqrt{t},\sqrt{t}] = P(X \leq \sqrt{t}) - P(X < \sqrt{t}) $$ Continuing this logic, and noting that $X$ has density $f_X = \frac{1}{\sqrt{2 \pi}} e^{-x^2/2}$, we see that $$ F_{X^2}(t) = F_X(\sqrt{t}) - F_X(-\sqrt{t}) = \displaystyle\int_{-\infty}^{\sqrt{t}} \frac{1}{\sqrt{2 \pi}} e^{-x^2/2} dx - \displaystyle\int_{-\infty} ^{-\sqrt{t}} \frac{1}{\sqrt{2 \pi}} e^{-x^2/2} dx$$ We can then find the density $f_{X^2}$ by differentiating our equation for $F_{X^2}(t)$ with respect to $t$, which gives us $$f_{X^2}(t) = \frac{d}{dt}\left(F_{X^2}(t) \right) = \frac{1}{\sqrt{2 \pi}} e^{- (\sqrt{t})^2 / 2} \left( \frac{1}{2 \sqrt{t}} \right) - \frac{1}{\sqrt{2 \pi}} e^{- (-\sqrt{t})^2 / 2} \left(- \frac{1}{2 \sqrt{t}} \right) $$ and this simplifies to tell us that $$f_{X^2}(t) = f_{Y^2}(t) = \frac{1}{\sqrt{t} \sqrt{2 \pi}} e^{- t/2} $$

Because we know that $X$ and $Y$ are independent, it follows that $X^2$ and $Y^2$ are independent as well. Therefore we can apply the convolution method to find the probability density for $Z = X^2 + Y^2$. We can write the probability distribution as the convolution $$f_Z(z) = \displaystyle\int_{0}^z f_{X^2}(z-y) f_{Y^2}(y) dy $$ where the lower-limit of the integral is 0 because $Y^2$ is a non-negative random variable, and the upper limit is $z$ because it cannot be the case that both $Y^2 = y > z$ and $X^2 < z - y$, as $X^2 $ is a non-negative random variable. Plugging our value for the densities $f_{X^2}(z-y) = \frac{1}{\sqrt{z-y} \sqrt{2 \pi}} e^{-(z-y)/2} $ and $f_{Y^2} = \frac{1}{\sqrt{y} \sqrt{2 \pi}} e^{-y/2}$, we have that $$f_Z(z) = \displaystyle\int_0^z \left( \frac{1}{\sqrt{z-y} \sqrt{2 \pi}} e^{-(z-y)/2} \right) \left( \frac{1}{\sqrt{y} \sqrt{2 \pi}} e^{-y/2} \right) dy = \frac{1}{2 \pi} e^{-z/2} \displaystyle\int_0^z \displaystyle\frac{dy}{\sqrt{zy - y^2}}$$

Now, what we have left to do is to compute the integral on the right. Completing the square under the square root sign we have $$\displaystyle\int_0^z \displaystyle\frac{dy}{\sqrt{zy - y^2}} = \displaystyle\int_0^z \displaystyle\frac{dy}{\sqrt{\frac{z^2}{4} - (\frac{z^2}{4} - zy + y^2) }} = \displaystyle\int_0^z \displaystyle\frac{dy}{\sqrt{\frac{z^2}{4} - (y - \frac{z}{2})^2 }} $$ Making the u-substitution $u = y - \frac{z}{2}$ (with $du = dy$ and with transformed limits of integration $y = 0 \mapsto u = -z/2$ and $y = z \mapsto z/2$), we obtain the integral $$ \displaystyle\int_0^z \displaystyle\frac{dy}{\sqrt{zy - y^2}} = \displaystyle\int_{-z/2}^{z/2} \displaystyle\frac{du}{\sqrt{\frac{z^2}{4} - u^2}} = \frac{2}{z} \displaystyle\int_{-z/2}^{z/2} \displaystyle\frac{du}{\sqrt{1 - (\frac{2u}{z})^2}} = \frac{2}{z} \left( \left(\frac{1}{\frac{2}{z}} \right) \sin^{-1}(\frac{2u}{z}) \bigg|_{-z/2}^{z/2} \right) $$ This simplifies to $$ \displaystyle\int_0^z \displaystyle\frac{dy}{\sqrt{zy - y^2}} = \sin^{-1}(1) - \sin^{-1}(-1) = \frac{\pi}{2} - (- \frac{\pi}{2}) = \pi $$ and therefore we have that $$f_{Z}(z) = \frac{1}{2 \pi} e^{-z/2} \left( \pi \right) \Longrightarrow \boxed{f_{Z}(z) = \frac{1}{2} e^{-z/2}} $$

And we can compute that $$P(X^2 + Y^2 <1 ) = P(Z < 1) = \displaystyle\int_0^1 f_{Z}(z)dz = \displaystyle\int_0^1 \frac{1}{2} e^{-z/2} dz = - e^{-z/2} \bigg|_0^1 $$ and we obtain our final answer $$ \boxed{P(X^2 + Y^2 < 1) = 1 - e^{-1/2}} $$

As per my knowledge you don't need to integrate , this type of questions are normally asked in competetive exam if you started integrating there then it will take much time. I have a shortcut.

$$X\sim N(0,1)$$

$$Y \sim N(0,1)$$

then

$$X^2 \sim\chi_1^2$$

$$Y^2 \sim\chi_1^2$$

$$\chi_1^2+\chi_1^2 \sim\chi_2^2$$

Chi-square distribution with $2$ degree of freedom will follow Gamma distribution with parameters $(1,\frac12)$ which is an exponential distribution with parameter $\lambda=\frac12$ so now we have an exponetial distribution lets call it $T$ so

$$ T \sim expo(1/2) $$

we have to find

$$P(T<1)$$

as we know $P(T<1)$ is cumulative ditribution function of exponetial ditribution which is generally given by

$$F(x) = 1- e^{-\lambda x}$$

as we know $\lambda$=1/2 $$F(1)= 1-e^{{-{1\over 2}}\times 1}$$

$$\Rightarrow P(X^2 + Y^2 < 1) = 1-e^{-1/2}$$