Evaluate {x} (x³+2x-3) dx

Correct Answer is : 2x9/29+ 4x5/25−2x3/2+c\;{2x^{9/2}}{9}+\;{4x^{5/2}}{5}-2x³/2}+c92x9/2+54x5/2−2x3/2+c

\;{x^{7/2}}{5}+\;{3x²}{7}-8x+c

\;{2x^{9/2}}{9}+\;{4x^{5/2}}{5}-2x³/2}+c

\;{3x^{7/2}}{9}+\;{x³/2}}{5}-2x³/2}+c

\;{2x^{5/2}}{7}-\;{x³/2}}{9}-2x^{5/2}+c

Correct Answer is : 2x9/29+ 4x5/25−2x3/2+c\;{2x^{9/2}}{9}+\;{4x^{5/2}}{5}-2x³/2}+c92x9/2+54x5/2−2x3/2+c

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Integral Calculus

Question : 44 of 73

Marks: +1, -0

Evaluate

\int \sqrt{x} (x^{3}+2x-3) dx

Solution:

Concept:

Simplify the integrand by multiplying

\sqrt{x}

with each term, then integrate using the power rule.

Explanation:

First, rewrite

\sqrt{x}

x^{1/2}

Multiply:

x^{1/2} \cdot x^3 = x^{7/2}

x^{1/2} \cdot 2x = 2x^{3/2}

x^{1/2} \cdot (-3) = -3x^{1/2}

So the integral becomes

\int (x^{7/2} + 2x^{3/2} - 3x^{1/2}) \, dx

Integrate each term:

\int x^{7/2} \, dx = \frac{x^{9/2}}{9/2} = \frac{2}{9}x^{9/2}

\int 2x^{3/2} \, dx = 2 \cdot \frac{x^{5/2}}{5/2} = \frac{4}{5}x^{5/2}

\int -3x^{1/2} \, dx = -3 \cdot \frac{x^{3/2}}{3/2} = -2x^{3/2}

Combine with constant of integration

c

\frac{2}{9}x^{9/2} + \frac{4}{5}x^{5/2} - 2x^{3/2} + c

Answer:

\frac{2x^{9/2}}{9} + \frac{4x^{5/2}}{5} - 2x^{3/2} + c

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