Math · Calculus I · Worked example

Handle a constant factor in u-substitution

Find ∫x²√(x³ + 1) dx.

\int x^{2} \sqrt{x^{3} + 1} d x

Choose u

The expression under the root is x³ + 1, and its derivative 3x² matches the factor x² up to the constant 3. Let u = x³ + 1.

u = x^{3} + 1

du = 3x² dx, so x² dx = du/3.

x^{2} d x = \frac{d u}{3}

The integral becomes (1/3)∫u^(1/2) du. Raise the power to 3/2 and divide by 3/2: (1/3)·(2/3)u^(3/2) = (2/9)u^(3/2).

\frac{1}{3} \int u^{1 / 2} d u = \frac{2}{9} u^{3 / 2} + C

Replace u with x³ + 1.

\frac{2}{9} {(x^{3} + 1)}^{3 / 2} + C

∫x²√(x³ + 1) dx = (2/9)(x³ + 1)^(3/2) + C.

Find ∫x/(x² + 1) dx.

Show the answer and explanation

Let u = x² + 1, so x dx = du/2. Then ∫(1/u)(du/2) = ½ ln|u| + C. Since x² + 1 is always positive, the absolute value can be dropped.

\begin{array}{l} F (x) = \frac{1}{2} ln (x^{2} + 1) \\ F^{'} (x) = \frac{x}{x^{2} + 1} \end{array}

Open the rows in Math and change 2/9 to 2/3, the answer you get by forgetting the 1/3 from du. The box marks it as not an antiderivative.

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