Solve ∫ [a(t)]^2/[a(t)]^2+[b(t)]^2dt

$$\int \frac { [ a ( t ) ] ^ { 2 } } { [ a ( t ) ] ^ { 2 } + [ b ( t ) ] ^ { 2 } } d t$$

Evaluate

$$\frac{ta^{2}}{a^{2}+b^{2}}+С$$

$$b\neq 0\text{ or }a\neq 0$$

Solution Steps

Expand \left(at\right)^{2}.

$$\int \frac{a^{2}t^{2}}{\left(at\right)^{2}+\left(bt\right)^{2}}\mathrm{d}t$$

Expand \left(at\right)^{2}.

$$\int \frac{a^{2}t^{2}}{a^{2}t^{2}+\left(bt\right)^{2}}\mathrm{d}t$$

Expand \left(bt\right)^{2}.

$$\int \frac{a^{2}t^{2}}{a^{2}t^{2}+b^{2}t^{2}}\mathrm{d}t$$

Factor the expressions that are not already factored in \frac{a^{2}t^{2}}{a^{2}t^{2}+b^{2}t^{2}}.

$$\int \frac{a^{2}t^{2}}{t^{2}\left(a^{2}+b^{2}\right)}\mathrm{d}t$$

Cancel out t^{2} in both numerator and denominator.

$$\int \frac{a^{2}}{a^{2}+b^{2}}\mathrm{d}t$$

Find the integral of \frac{a^{2}}{a^{2}+b^{2}} using the table of common integrals rule \int a\mathrm{d}t=at.

$$\frac{a^{2}}{a^{2}+b^{2}}t$$

Simplify.

$$\frac{a^{2}t}{a^{2}+b^{2}}$$

If F\left(t\right) is an antiderivative of f\left(t\right), then the set of all antiderivatives of f\left(t\right) is given by F\left(t\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\begin{matrix}\frac{a^{2}t}{a^{2}+b^{2}}+С_{5},&\end{matrix}$$

Differentiate w.r.t. a

$$2\times \left(\frac{b}{a^{2}+b^{2}}\right)^{2}at$$