$\"\"$

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The integral is $\"\"$ .

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Find the integral using integration by parts.

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Integration by parts formula: $\"\"$ .

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Let $\"\"$ and $\"\"$ .

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Find $\"\"$ by integrating $\"\"$ .

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$\"\"$

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$\"\"$ .

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Differentiate on each side.

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$\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ in integration by parts formula.

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$\"\"$ $\"\"$

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Consider $\"\"$ .

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Find the integral using integration by parts.

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Integration by parts formula: $\"\"$ .

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Let $\"\"$ and $\"\"$ .

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Find $\"\"$ by integrating $\"\"$ .

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$\"\"$

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$\"\"$ .

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Differentiate on each side.

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$\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ in integration by parts formula.

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$\"\"$

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Substitute above result in $\"\"$ .

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$\"\"$

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$\"\"$

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Consider the antiderivative function by taking $\"\"$ as $\"\"$ .

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Graph the integrand function $\"\"$ and its antiderivative $\"\"$ .

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$\"\"$

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Observe the graph,

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When $\"\"$ is negative, then $\"\"$ is decreasing and $\"\"$ is positive , then $\"\"$ is increasing.

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Hence the curve $\"\"$ is derivative of the curve $\"\"$ .

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Therefore, $\"\"$ is antiderivative of the integrand function $\"\"$ .

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$\"\"$

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$\"\"$ .

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Graph:

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$\"\"$