In \(\int 6x(3x^2+4)^5\,dx\), what substitution makes the inner expression a single variable?

For \(u=5x-1\), write \(du\) in terms of \(dx\).

Evaluate \(\int 4x(x^2+7)^3\,dx\).

Evaluate \(\int 3\cos(3x+2)\,dx\).

Evaluate \(\int \frac{2x}{x^2+9}\,dx\).

Evaluate \(\int x\sqrt{x^2+1}\,dx\).

Evaluate \(\int (2x-5)e^{x^2-5x}\,dx\).

Evaluate \(\int \frac{x^2}{1+x^3}\,dx\).

Evaluate \(\int_0^2 2x(x^2+1)^4\,dx\) by changing the limits.

Evaluate \(\int \frac{\sin(\sqrt{x})}{\sqrt{x}}\,dx\) for \(x>0\).

Evaluate \(\int x(4-x^2)^6\,dx\).

Evaluate \(\int_1^e \frac{\ln x}{x}\,dx\), and explain why the substitution changes both limits.

Evaluate \(\int x^3\sqrt{x^2+2}\,dx\).

Evaluate \(\int \frac{x}{(x+1)^4}\,dx\).

For which constant \(k\) does \(\int (kx+3)(x^2+3x+8)^7\,dx\) become a direct single-substitution integral using \(u=x^2+3x+8\)? Then evaluate it.

Evaluate \(\int \frac{x+1}{x^2+2x+5}\,dx\), making any completion of the square explicit.

Evaluate \(\int_0^1 \frac{x}{(1+x^2)^2}\,dx\), and state whether the value is positive or negative before computing it.

A student claims \(\int 2x\cos(x^2)\,dx=2x\sin(x^2)+C\). Identify the error and give the correct antiderivative.

Evaluate \(\int \frac{x^2}{\sqrt{1+x^3}}\,dx\) and justify the constant factor carefully.

Show that \(\int_a^b f'(x)F(f(x))\,dx=\int_{f(a)}^{f(b)}F(u)\,du\), assuming \(f\) is differentiable and the substitution is valid.