Worked Examples: Techniques of Integration

Evaluate $\int 2x(x^2+1)^5\,dx$.

Evaluate $\int e^x\sin x\,dx$.

Evaluate $\int xe^x\,dx$.

Use the tabular method to evaluate $\int x^3e^{2x}\,dx$.

Evaluate $\int\sin^3x\,dx$.

Evaluate $\int\tan^3x\sec^4x\,dx$.

Explain why the rule “odd tangent power means save $\sec x\tan x$” cannot be used for $\int\tan^3x\,dx$ as written.

Evaluate $\int\frac{dx}{x^2\sqrt{4-x^2}}$ on an interval contained in $0<x<2$.

For $x\ge a>0$, simplify $\sqrt{x^2-a^2}$ using $x=a\sec\theta$.

Evaluate $\int\frac{1}{x^2-1}\,dx$.

Evaluate $\int\frac{2x^2-x+4}{x^3+4x}\,dx$.

Use Wallis' formula to evaluate $\int_0^{\pi/2}\sin^6x\,dx$.