∫xe^x dx के लिए चार विकल्प दिए गए हैं। यदि A=[2+10−i0] और B=[1-i2i0] का मान बताओ। tan^(-1)(√3)−sec^−1(−2) का मान ज्ञात कीजिये। ∫xe^x dx के लिए चार विकल्प दिए गए हैं। यदि A=[2+10−i0] और B=[1-i2i0] का मान बताओ। tan^(-1)(√3)−sec^−1(−2) का मान ज्ञात कीजिये। Read more

Steps to Solve

1. SIntegral Calculation of $\int xe^x dx$

To solve $\int xe^x dx$, we can use integration by parts. Let:

• $u = x$
• $dv = e^x dx$

Then, we find:

• $du = dx$
• $v = e^x$

Using the integration by parts formula:

$$ \int u , dv = uv - \int v , du $$

This gives:

$$ \int xe^x dx = x e^x - \int e^x dx $$

Now, calculating:

$$ \int e^x dx = e^x $$

So we have:

$$ \int xe^x dx = x e^x - e^x + C = (x - 1)e^x + C $$

2. Identify the Correct Option

From the options given:

1. $e^x$
2. $(x + 1)e^x$
3. $(x - 1)e^x$
4. $\frac{x^2}{2} e^x$

The correct answer is option (3) $(x - 1)e^x$.

3. Matrix Addition $(A + B)$

Given matrices:

$$ A = \begin{bmatrix} 2 + 10i & 0 \ 3 & 4i \end{bmatrix}, \quad B = \begin{bmatrix} 1 - i & 2i \ 0 & 3 \end{bmatrix} $$

To find $A + B$, we add corresponding elements:

$$ A + B = \begin{bmatrix} (2 + 10i) + (1 - i) & 0 + 2i \ 3 + 0 & 4i + 3 \end{bmatrix} $$

Calculating the elements:

• First element: $(2 + 10i) + (1 - i) = 3 + 9i$
• Second element: $0 + 2i = 2i$
• Third element: $3 + 0 = 3$
• Fourth element: $4i + 3 = 3 + 4i$

So,

$$ A + B = \begin{bmatrix} 3 + 9i & 2i \ 3 & 3 + 4i \end{bmatrix} $$

4. Calculate $tan^{-1}(\sqrt{3}) - sec^{-1}(-2)$

Calculating these values separately:

• $tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$ (because $\tan \frac{\pi}{3} = \sqrt{3}$)

• For $sec^{-1}(-2)$, we find: $$ sec^{-1}(-2) = \pi - sec^{-1}(2) $$ Since $sec^{-1}(2) = \frac{\pi}{3}$, thus: $$ sec^{-1}(-2) = \pi - \frac{\pi}{3} = \frac{2\pi}{3} $$

Putting it all together:

$$ tan^{-1}(\sqrt{3}) - sec^{-1}(-2) = \frac{\pi}{3} - \frac{2\pi}{3} = -\frac{\pi}{3} $$