Integration by Substitution

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A particle moves along a line with velocity $v(t) = t\sqrt{t^2+1}$ m/s. Find its displacement function $s(t) = \int v(t)\,dt$ (indefinite integral).

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

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

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

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

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Evaluate the indefinite integral: $\int \frac{2x+1}{\sqrt{x^2+x+1}} \, dx$

\sqrt{x^2+x+1} + C

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

\frac{1}{2}\sqrt{x^2+x+1} + C

2\sqrt{x^2+x+1} + C

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The electric field along a line is proportional to $\frac{x^2+1}{x^4+1}$ . Find its potential function (indefinite integral) $\int \frac{x^2+1}{x^4+1} \, dx$ .

\frac{1}{2} \ln|x^4+1| + C

\frac{1}{2} \arctan(x^2) + C

\frac{1}{\sqrt{2}} \arctan\left(\frac{x^2-1}{\sqrt{2}\,x}\right) + C

\frac{1}{\sqrt{2}} \arctan\left(\frac{x^2}{\sqrt{2}}\right) + C

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Integration by Substitution

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