Definite Integrals

Answer

We use the formula: $$ \int_a^b f(x)\,dx = \lim_{n \to \infty} \frac{b-a}{n} \sum_{k=0}^{n-1} f\left(a + k\cdot h\right), \quad h = \frac{b-a}{n} $$ Here, $a = 1$, $b = 2$, $f(x) = x$, $h = \frac{1}…

Answer

Using the limit of sum: $$ \int_a^b f(x)\,dx = \lim_{h \to 0} h \sum_{k=0}^{n-1} f(a + kh), \quad h = \frac{b-a}{n} $$ Here, $a = 0$, $b = 2$, $f(x) = e^x$, $h = \frac{2}{n}$ $$ \int_0^2 e^x\,dx = …

Answer

Step 1: Find the antiderivative of $f(x) = x$: $$ \int x\,dx = \frac{x^2}{2} + C $$ Step 2: Apply the limits: $$ \int_1^2 x\,dx = \left[ \frac{x^2}{2} \right]_1^2 = \frac{2^2}{2} - \frac{1^2}{2} = …

Answer

Step 1: Antiderivative of $\cos x$ is $\sin x$: $$ \int \cos x\,dx = \sin x + C $$ Step 2: Apply limits: $$ \int_0^{\pi/2} \cos x\,dx = \left[ \sin x \right]_0^{\pi/2} = \sin\left(\frac{\pi}{2}\rig…

Answer

Step 1: Antiderivative of $e^{2x}$: $$ \int e^{2x}\,dx = \frac{e^{2x}}{2} + C $$ Step 2: Apply limits: $$ \int_0^2 e^{2x}\,dx = \left[ \frac{e^{2x}}{2} \right]_0^2 = \frac{e^4}{2} - \frac{e^0}{2} =…

Answer

Let $u = 1 + x^2$, then $du = 2x\,dx \Rightarrow x\,dx = \frac{1}{2}du$ When $x = 2$, $u = 5$; when $x = 3$, $u = 10$ $$ \int_2^3 \frac{x}{1+x^2}\,dx = \frac{1}{2} \int_5^{10} \frac{1}{u}\,du = \fra…

Answer

Let $u = \cos x$, then $du = -\sin x\,dx$ When $x = 0$, $u = 1$; when $x = \pi/2$, $u = 0$ $$ \int_0^{\pi/2} \frac{\sin x}{1 + \cos^2 x}\,dx = -\int_1^0 \frac{1}{1 + u^2}\,du = \int_0^1 \frac{1}{1 +…

Answer

Use identity: $\cos x = \frac{1 - \tan^2(x/2)}{1 + \tan^2(x/2)}$ Let $t = \tan(x/2)$, then $\cos x = \frac{1 - t^2}{1 + t^2}$, $dx = \frac{2\,dt}{1 + t^2}$ When $x = 0$, $t = 0$; when $x = \pi/2$, $…