Advertisements
Advertisements
Question
Advertisements
Solution
\[Let\ I = \int_0^7 \frac{\sqrt[3]{x}}{\sqrt[3]{x} + \sqrt[3]{7 - x}} d x ..................(1)\]
\[ = \int_0^7 \frac{\sqrt[3]{7 - x}}{\sqrt[3]{7 - x} + \sqrt[3]{x}} dx .................\left(\text{Using }\int_0^a f\left( x \right) dx = \int_0^a f\left( a - x \right) dx \right)\]
\[ = \int_0^7 \frac{\sqrt[3]{7 - x}}{\sqrt[3]{x} + \sqrt[3]{7 - x}} dx ..................(2)\]
\[\text{Adding (1) and (2) we get}\]
\[2I = \int_0^7 \frac{\sqrt[3]{x} + \sqrt[3]{7 - x}}{\sqrt[3]{x} + \sqrt[3]{7 - x}} d x \]
\[ = \int_0^7 dx\]
\[ = \left[ x \right]_0^7 = 7\]
\[Hence\ I = \frac{7}{2}\]
APPEARS IN
RELATED QUESTIONS
Evaluate the following definite integrals:
\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]
If \[f\left( x \right) = \int_0^x t\sin tdt\], the write the value of \[f'\left( x \right)\]
`int_0^1 sqrt((1 - "x")/(1 + "x")) "dx"`
If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\] then the value of I10 + 90I8 is
Evaluate : \[\int\limits_0^\pi/4 \frac{\sin x + \cos x}{16 + 9 \sin 2x}dx\] .
\[\int\limits_0^4 x\sqrt{4 - x} dx\]
\[\int\limits_0^{\pi/4} \tan^4 x dx\]
\[\int\limits_0^{\pi/2} \left| \sin x - \cos x \right| dx\]
\[\int\limits_{- a}^a \frac{x e^{x^2}}{1 + x^2} dx\]
\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx\]
\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]
\[\int\limits_0^{\pi/2} \frac{1}{2 \cos x + 4 \sin x} dx\]
\[\int\limits_0^2 \left( 2 x^2 + 3 \right) dx\]
\[\int\limits_2^3 e^{- x} dx\]
Using second fundamental theorem, evaluate the following:
`int_(-1)^1 (2x + 3)/(x^2 + 3x + 7) "d"x`
Evaluate `int sqrt((1 + x)/(1 - x)) "d"x`, x ≠1
`int (x + 3)/(x + 4)^2 "e"^x "d"x` = ______.
