Integration techniques

Evaluate $\int\dfrac1{2x}(\ln x)^5\,dx$.

Without the use of a calculator, show that

(a) Find $\int\dfrac1{3-4t^2}dt$.

(a) Write down $\dfrac d{dx}(\tan(x^2))$. Hence find $\int x^3\sec^2(x^2)dx$.

(a) Find $\int x\cos x^2 dx$.

(i) Use the substitution $u=2x-1$ to find $\int\dfrac x{\sqrt{1-(2x-1)^2}}dx$.

Find $\int e^{2x}\cos3x dx$.

(a) Given that $A(\sin\theta+\cos\theta)+B(\cos\theta-\sin\theta)\equiv4\sin\theta$, find the constants $A,B$.

(a) Use $u=e^x+2$ to find the exact value of $\int_0^{\ln2}\dfrac{e^{3x}}{e^x+2}dx$.

(a) Use $x=2\tan\theta$ to find the exact value of $\int_{2/\sqrt3}^{2}\dfrac1{x\sqrt{x^2+4}}dx$.

(a)(i) Find $\int\dfrac{2-x}{\sqrt{4x-x^2-1}}dx$.

(a) Use $x=\tan t$, where $0<t<\pi/2$, to find $\int\dfrac1{x^2\sqrt{x^2+1}}dx$.

(i) Verify without a calculator that $\theta=\pi/4$ is a root of $\tan^2\theta-2\cos^2\theta=0$.

(i) Find $\dfrac d{dx}(\sin^{-1}x^2)$.