1 ∫ 0 X E X 2 D X - Mathematics

प्रश्न

\[\int\limits_0^1 x e^{x^2} dx\]

उत्तर

\[Let\ I = \int_0^1 x\ e^{x^2} d\ x . \]
\[Let\ x^2 = t . Then, 2x\ dx = dt\]
\[When\ x = 0, t = 0\ and\ x\ = 1\, t = 1\]
\[ \therefore I = \frac{1}{2} \int_0^1 e^t\ dt\]
\[ \Rightarrow I = \frac{1}{2} \left( e^t \right)_0^1 \]
\[ \Rightarrow I = \frac{1}{2}\left( e - 1 \right)\]

shaalaa.com

Definite Integrals

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 7 Q 6 Q 8

अध्याय 20: Definite Integrals - Exercise 20.2 [पृष्ठ ३८]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12

अध्याय 20 Definite Integrals
Exercise 20.2 | Q 7 | पृष्ठ ३८

संबंधित प्रश्न

\[\int\limits_0^\infty e^{- x} dx\]

\[\int\limits_0^{\pi/2} \left( \sin x + \cos x \right) dx\]

\[\int\limits_0^{\pi/2} \left( a^2 \cos^2 x + b^2 \sin^2 x \right) dx\]

\[\int\limits_0^{\pi/2} \sqrt{1 + \sin x}\ dx\]

\[\int\limits_0^{\pi/2} \sqrt{1 + \cos x}\ dx\]

\[\int\limits_0^{\pi/2} \sin^3 x\ dx\]

\[\int\limits_0^a \frac{x}{\sqrt{a^2 + x^2}} dx\]

\[\int\limits_0^a \sqrt{a^2 - x^2} dx\]

\[\int\limits_0^1 \frac{\sqrt{\tan^{- 1} x}}{1 + x^2} dx\]

\[\int\limits_0^{\pi/2} \frac{\sin x \cos x}{1 + \sin^4 x} dx\]

\[\int\limits_0^{\pi/2} \frac{1}{5 + 4 \sin x} dx\]

\[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\]

\[\int\limits_0^{\pi/2} \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} dx\]

\[\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{3/2}} dx\]

\[\int_0^\frac{\pi}{2} \frac{\cos x}{\left( \cos\frac{x}{2} + \sin\frac{x}{2} \right)^n}dx\]

\[\int\limits_0^9 f\left( x \right) dx, where f\left( x \right) \begin{cases}\sin x & , & 0 \leq x \leq \pi/2 \\ 1 & , & \pi/2 \leq x \leq 3 \\ e^{x - 3} & , & 3 \leq x \leq 9\end{cases}\]

\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]

\[\int_{- 1}^2 \left( \left| x + 1 \right| + \left| x \right| + \left| x - 1 \right| \right)dx\]

Evaluate each of the following integral:

\[\int_0^{2\pi} \log\left( \sec x + \tan x \right)dx\]

\[\int\limits_0^a \frac{1}{x + \sqrt{a^2 - x^2}} dx\]

Evaluate the following integral:

\[\int_{- a}^a \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right)d\theta\]

If f(2a − x) = −f(x), prove that

\[\int\limits_0^{2a} f\left( x \right) dx = 0 .\]

If f is an integrable function, show that

\[\int\limits_{- a}^a x f\left( x^2 \right) dx = 0\]

\[\int\limits_1^2 \left( x^2 - 1 \right) dx\]

\[\int\limits_a^b e^x dx\]