1 Integration Techniques Online Quiz Questions

$u=6x$

$u=3x^2+4$

$u=(3x^2+4)^4$

$u=x^2$

$x=a\sin\theta$

$x=a\sec\theta$

$x=a\cos\theta$

$x=a\tan\theta$

$\int_a^a f(x)\,dx=0$

$\int_a^b f(x)\,dx=\int_b^a f(x)\,dx$

$\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx$

$\int_a^c f(x)\,dx-\int_c^b f(x)\,dx=\int_a^b f(x)\,dx$

$\int_a^c f(x)\,dx+\int_c^b f(x)\,dx=\int_a^b f(x)\,dx$

Use substitution for $\int 2x\cos(x^2)\,dx$.

Use integration by parts for $\int xe^x\,dx$.

Use partial fractions for $\int \sin(x^2)\,dx$.

Use partial fractions for $\int \frac{5x+1}{(x-1)(x+2)}\,dx$.

Use trigonometric substitution for $\int \frac{dx}{\sqrt{9+x^2}}$.

True

False

True

False

$u=x,\;dv=\ln x\,dx$

$u=1,\;dv=x\ln x\,dx$

$u=\ln x,\;dv=x\,dx$

$u=x\ln x,\;dv=dx$

It must include an arbitrary constant because every antiderivative contains one.

It does not include an arbitrary constant because endpoint subtraction cancels the constant.

It always represents geometric area, even where the function is negative.

It can be evaluated only by constructing a limit of Riemann sums.

$\frac{(3x^2+4)^4}{4}+C$

$6(3x^2+4)^5+C$

$\frac{(3x^2+4)^5}{5}+C$

$\frac{(3x^2+4)^6}{6}+C$

True

False

$x\sin x+\cos x+C$

$x\cos x-\sin x+C$

$x\sin x-\cos x+C$

$\sin x+x\cos x+C$

Use $u=\sin x$ after converting every cosine factor to a sine factor.

Save one factor of $\sin x$, convert the remaining sine power using $\sin^2x=1-\cos^2x$, and let $u=\cos x$.

Apply the half-angle identity to both powers immediately because both exponents are even.

Save one factor of $\cos x$ and let $u=\sin x$, because the cosine exponent is even.

$\frac{A}{(x-a)^3}$

$\frac{A}{x-a}+\frac{B}{(x-a)^3}$

$\frac{Ax+B}{(x-a)^3}$

$\frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{(x-a)^3}$