Solve ∫ (1-t^2)^3tdt

$$\int ( 1 - t ^ { 2 } ) ^ { 3 } t d t$$

Evaluate

$$-\frac{t^{8}}{8}+\frac{t^{6}}{2}-\frac{3t^{4}}{4}+\frac{t^{2}}{2}+С$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(1-t^{2}\right)^{3}.

$$\int \left(1-3t^{2}+3\left(t^{2}\right)^{2}-\left(t^{2}\right)^{3}\right)t\mathrm{d}t$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\int \left(1-3t^{2}+3t^{4}-\left(t^{2}\right)^{3}\right)t\mathrm{d}t$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\int \left(1-3t^{2}+3t^{4}-t^{6}\right)t\mathrm{d}t$$

Use the distributive property to multiply 1-3t^{2}+3t^{4}-t^{6} by t.

$$\int t-3t^{3}+3t^{5}-t^{7}\mathrm{d}t$$

Integrate the sum term by term.

$$\int t\mathrm{d}t+\int -3t^{3}\mathrm{d}t+\int 3t^{5}\mathrm{d}t+\int -t^{7}\mathrm{d}t$$

Factor out the constant in each of the terms.

$$\int t\mathrm{d}t-3\int t^{3}\mathrm{d}t+3\int t^{5}\mathrm{d}t-\int t^{7}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t\mathrm{d}t with \frac{t^{2}}{2}.

$$\frac{t^{2}}{2}-3\int t^{3}\mathrm{d}t+3\int t^{5}\mathrm{d}t-\int t^{7}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{3}\mathrm{d}t with \frac{t^{4}}{4}. Multiply -3 times \frac{t^{4}}{4}.

$$\frac{t^{2}}{2}-\frac{3t^{4}}{4}+3\int t^{5}\mathrm{d}t-\int t^{7}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{5}\mathrm{d}t with \frac{t^{6}}{6}. Multiply 3 times \frac{t^{6}}{6}.

$$\frac{t^{2}}{2}-\frac{3t^{4}}{4}+\frac{t^{6}}{2}-\int t^{7}\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{7}\mathrm{d}t with \frac{t^{8}}{8}. Multiply -1 times \frac{t^{8}}{8}.

$$\frac{t^{2}}{2}-\frac{3t^{4}}{4}+\frac{t^{6}}{2}-\frac{t^{8}}{8}$$

Simplify.

$$-\frac{t^{8}}{8}+\frac{t^{6}}{2}-\frac{3t^{4}}{4}+\frac{t^{2}}{2}$$

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.

$$-\frac{t^{8}}{8}+\frac{t^{6}}{2}-\frac{3t^{4}}{4}+\frac{t^{2}}{2}+С$$

Differentiate w.r.t. t

$$t\left(-t-1\right)\left(t+1\right)^{2}\left(t-1\right)^{3}$$