Basic indefinite integration

5 \includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-06_355_634_255_753} The diagram shows a semicircle with diameter \(A B\), centre \(O\) and radius \(r\). The point \(C\) lies on the circumference and angle \(A O C = \theta\) radians. The perimeter of sector \(B O C\) is twice the perimeter of sector \(A O C\). Find the value of \(\theta\) correct to 2 significant figures.

Simplify the following expressions fully.

1. \(\left( x ^ { 6 } \right) ^ { \frac { 1 } { 3 } }\)
2. \(\sqrt { 2 } \left( x ^ { 3 } \right) \div \sqrt { \frac { 32 } { x ^ { 2 } } }\)

Find \(\int \left( 3 x ^ { 2 } + 4 x ^ { 5 } - 7 \right) d x\).

Find \(\int \left( 6 x ^ { 2 } + 2 + x ^ { - \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving each term in its simplest form.

2. Find $$\int \left( 8 x ^ { 3 } + 6 x ^ { \frac { 1 } { 2 } } - 5 \right) d x$$ giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-03_40_38_2682_1914}

1. Find

$$\int \left( 6 x ^ { 2 } + \frac { 2 } { x ^ { 2 } } + 5 \right) \mathrm { d } x$$ giving each term in its simplest form.

2. Find $$\int \left( 10 x ^ { 4 } - 4 x - \frac { 3 } { \sqrt { } x } \right) \mathrm { d } x$$ giving each term in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-03_120_51_2599_1900}

1. Find

$$\int \left( 2 x ^ { 4 } - \frac { 4 } { \sqrt { } x } + 3 \right) d x$$ giving each term in its simplest form.

1. Find

$$\int \left( 2 x ^ { 5 } - \frac { 1 } { 4 x ^ { 3 } } - 5 \right) \mathrm { d } x$$ giving each term in its simplest form.

Find $$\int \left( 12 x ^ { 5 } - 3 x ^ { 2 } + 4 x ^ { \frac { 1 } { 3 } } \right) \mathrm { d } x$$ giving each term in its simplest form.

1 Find

1. \(\int \left( x ^ { 3 } + 8 x - 5 \right) \mathrm { d } x\),
2. \(\int 12 \sqrt { x } \mathrm {~d} x\).

4 Show that \(\sum _ { r = 1 } ^ { 4 } \ln \frac { r } { r + 1 } = - \ln 5\).

Find \(\int \left(x^3 + \frac{1}{x^3}\right) \mathrm{d}x\). [3]

Find \(\int (6x^2 + 2x + x^{-2}) \, dx\), giving each term in its simplest form. [4]

\(y = 7 + 10x^{\frac{3}{2}}\).

1. Find \(\frac{dy}{dx}\). [2]
2. Find \(\int y \, dx\). [3]

Find \(\int 5x + 3\sqrt{x} \, dx\) [4]

\(y = 7 + 10x^{\frac{1}{3}}\).

1. Find \(\frac{dy}{dx}\). [2]
2. Find \(\int y \, dx\). [3]

Find $$\int \left( 3x^2 + \frac{1}{2x^2} \right) dx.$$ [4]

Find \(\int \left(x - \frac{3}{x^2}\right) dx\). [3]

Find \(\int 30x^2 dx\). [3]

Find \(\int 7x^3 \, dx\). [3]

Find \(\int 7x^2 dx\). [3]

Find \(\int 30x^2 dx\). [3]

Find \(\int (x^5 + 10x^3) dx\). [4]

Find \(\int (3x^5 + 2x^{-\frac{1}{2}}) dx\). [4]