CUET UG 2025 — Mathematics Calculus

The integral can be split into two parts:

$$\int_0^8 (x^{\frac{2}{3}} + 1) dx = \int_0^8 x^{\frac{2}{3}} dx + \int_0^8 1 dx$$∫08​(x32​+1)dx=∫08​x32​dx+∫08​1dx

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Using the power rule with $n = \frac{2}{3}$n=32​:

$$n + 1 = \frac{2}{3} + 1 = \frac{5}{3}$$n+1=32​+1=35​

$$\int x^{\frac{2}{3}} dx = \frac{x^{\frac{5}{3}}}{\frac{5}{3}} = \frac{3}{5}x^{\frac{5}{3}}$$∫x32​dx=35​x35​​=53​x35​

For the constant:

$$\int 1 dx = x$$∫1dx=x

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Combining the results:

$$\int_0^8 (x^{\frac{2}{3}} + 1) dx = \left[\frac{3}{5}x^{\frac{5}{3}} + x\right]_0^8$$∫08​(x32​+1)dx=[53​x35​+x]08​

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Evaluating $8^{\frac{5}{3}}$835​:

$$8^{\frac{5}{3}} = (8^{\frac{1}{3}})^5$$835​=(831​)5

$$8^{\frac{1}{3}} = 2$$831​=2

$$(2)^5 = 32$$(2)5=32

At $x = 8$x=8:

$$\frac{3}{5}(32) + 8 = \frac{96}{5} + \frac{40}{5} = \frac{136}{5}$$53​(32)+8=596​+540​=5136​

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At $x = 0$x=0:

$$\frac{3}{5}(0)^{\frac{5}{3}} + 0 = 0$$53​(0)35​+0=0

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The final result:

$$\frac{136}{5} - 0 = \frac{136}{5}$$5136​−0=5136​

Therefore, $\displaystyle \int_0^8 (x^{\frac{2}{3}} + 1) dx = \dfrac{136}{5}$∫08​(x32​+1)dx=5136​