Standard Integrals and Reverse Chain Rule

2 The function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 7\) and \(\mathrm { f } ( 3 ) = 5\). Find \(\mathrm { f } ( x )\).

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 2 } - 5\). Given that the point \(( 3,8 )\) lies on the curve, find the equation of the curve.

9 A curve which passes through \(( 0,3 )\) has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 2 } { ( x - 1 ) ^ { 3 } }\).

1. Find the equation of the curve.
   The tangent to the curve at \(( 0,3 )\) intersects the curve again at one other point, \(P\).
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(( 2 x + 1 ) ( x - 1 ) ^ { 2 } - 1 = 0\).
3. Verify that \(x = \frac { 3 } { 2 }\) satisfies this equation and hence find the \(y\)-coordinate of \(P\).

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

1 Find \(\int 12 x ^ { 3 } \mathrm {~d} x\)
Circle your answer.
\(36 x ^ { 2 } + c\)
\(3 x ^ { 4 } + c\)
\(3 x ^ { 2 } + c\)
\(36 x ^ { 4 } + c\)

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

2 Use integration to find the exact value of \(\int _ { \frac { 1 } { 16 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( 9 - 6 \cos ^ { 2 } 4 x \right) \mathrm { d } x\).

3

1. Find \(\int 4 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta\).
2. Find the exact value of \(\int _ { - 1 } ^ { 6 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x\).

5. \begin{figure}[h] \end{figure} Show that the area of the finite region between the curves \(y = \tan ^ { 2 } x\) and \(y = 4 \cos 2 x - 1\) in the interval \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), shown shaded in Figure 3, is given by $$2 \sqrt { 2 \sqrt { 3 } } - 2 \sqrt { 2 \sqrt { 3 } - 3 }$$

1 Show that \(\int _ { 2 } ^ { 8 } \frac { 3 } { x } \mathrm {~d} x = \ln 64\).

1 Show that \(\int _ { 2 } ^ { 8 } \frac { 3 } { x } \mathrm {~d} x = \ln 64\).

2 Fig. 7 shows the curve \(y = \frac { x ^ { 2 } } { 1 + 2 x ^ { 3 } }\). It is undefined at \(x = a\); the line \(x = a\) is a vertical asymptote. \begin{figure}[h] \end{figure}

1. Calculate the value of \(a\), giving your answer correct to 3 significant figures.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x - 2 x ^ { 4 } } { \left( 1 + 2 x ^ { 3 } \right) ^ { 2 } }\). Hence determine the coordinates of the turning points of the curve.
3. Show that the area of the region between the curve and the \(x\)-axis from \(x = 0\) to \(x = 1\) is \(\frac { 1 } { 6 } \ln 3\).

1 Find \(\int \left( x + \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\).

1 Find \(\int \left( x + \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\).

1. The curve \(C\) has equation

$$y = 2 + \ln \left( 1 - x ^ { 2 } \right) \quad \frac { 1 } { 2 } \leqslant x \leqslant \frac { 3 } { 4 }$$

1. Show that the length of the curve \(C\) is given by $$\int _ { \frac { 1 } { 2 } } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x$$
2. Hence, using algebraic integration, show that the length of the curve \(C\) is \(p + \ln q\) where \(p\) and \(q\) are rational numbers to be determined.

4 The equation of a curve is \(y = x ^ { 4 } + 4 x + 9\).

1. Find the coordinates of the stationary point on the curve and determine its nature.
2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\).

4 The equation of a curve is \(y = x ^ { 4 } + 4 x + 9\).

1. Find the coordinates of the stationary point on the curve and determine its nature.
2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\).

6．（a）Starting from \([ \mathrm { f } ( x ) - \lambda \mathrm { g } ( x ) ] ^ { 2 } \geqslant 0\) show that \(\lambda\) satisfies the quadratic inequality $$\left( \int _ { a } ^ { b } [ \operatorname { g } ( x ) ] ^ { 2 } \mathrm {~d} x \right) \lambda ^ { 2 } - 2 \left( \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right) \lambda + \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \geqslant 0$$ where \(a\) and \(b\) are constants and \(\lambda\) can take any real value．
（2）
（b）Hence prove that $$\left[ \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right] ^ { 2 } \leqslant \left[ \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \right] \times \left[ \int _ { a } ^ { b } [ \mathrm {~g} ( x ) ] ^ { 2 } \mathrm {~d} x \right]$$ （c）By letting \(\mathrm { f } ( x ) = 1\) and \(\mathrm { g } ( x ) = \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } }\) show that $$\int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x \leqslant \frac { 9 } { 2 }$$ （d）Show that \(\int _ { - 1 } ^ { 2 } x ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 4 } } \mathrm {~d} x = \frac { 12 \sqrt { } 3 } { 5 }\)
（e）Hence show that $$\frac { 144 } { 55 } \leqslant \int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x$$

1 Find the exact value of the constant \(k\) for which \(\int _ { 1 } ^ { k } \frac { 1 } { 2 x - 1 } \mathrm {~d} x = 1\).

2 A curve with equation \(y = \mathrm { f } ( x )\) passes through the points \(( 0,2 )\) and \(( 3 , - 1 )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = k x ^ { 2 } - 2 x\), where \(k\) is a constant. Find the value of \(k\).

10 A curve has equation \(y = \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } }\) where \(x > 0\) and \(k\) is a positive constant.

1. It is given that when \(x = \frac { 1 } { 4 }\), the gradient of the curve is 3 . Find the value of \(k\).
2. It is given instead that \(\int _ { \frac { 1 } { 4 } k ^ { 2 } } ^ { k ^ { 2 } } \left( \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } } \right) \mathrm { d } x = \frac { 13 } { 12 }\). Find the value of \(k\).

3. Evaluate $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin 2 x \cos x d x$$

1 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \cos 3 x \mathrm {~d} x\).

15. In this question you must use detailed reasoning.

1. Show that \(\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \frac { 1 + \sin 2 x } { - \cos 2 x } d x = \ln ( \sqrt { a } + b )\), where \(a\) and \(b\) are integers to be determined.
2. Show that \(\frac { \pi } { \frac { \pi } { 4 } } \frac { 1 + \sin 2 x } { - \cos 2 x } d x\) is undefined, explaining your reasoning clearly.
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11.

1. Given that \(u = 2 ^ { x }\), write down an expression for \(\frac { \mathrm { d } u } { \mathrm {~d} x }\)
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2. Find the exact value of \(\int _ { 0 } ^ { 1 } 2 ^ { x } \sqrt { 3 + 2 ^ { x } } \mathrm {~d} x\) Fully justify your answer.
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1 Find the exact value of \(\int _ { - 1 } ^ { 2 } \left( 4 \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x } \right) \mathrm { d } x\).

5 A curve has an equation which satisfies \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 3 x ^ { - \frac { 1 } { 2 } }\) for all positive values of \(x\). The point \(P ( 4,1 )\) lies on the curve, and the gradient of the curve at \(P\) is 5 . Find the equation of the curve.

3

1. Find the quotient when \(3 x ^ { 3 } - x ^ { 2 } + 10 x - 3\) is divided by \(x ^ { 2 } + 3\), and show that the remainder is \(x\).
2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } - x ^ { 2 } + 10 x - 3 } { x ^ { 2 } + 3 } \mathrm {~d} x$$

7 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 - x ^ { 2 } - 6 x\) passes through the point \(( 3 , - 10 )\).

1. Find the equation of the curve.
2. Express \(7 - x ^ { 2 } - 6 x\) in the form \(a - ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
3. Find the set of values of \(x\) for which the gradient of the curve is positive.

1 Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 2 } { x ^ { 2 } } d x\).

6 A curve passes through the point \(\left( \frac { 4 } { 5 } , - 3 \right)\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 20 } { ( 5 x - 3 ) ^ { 2 } }\).

1. Find the equation of the curve.
2. The curve is transformed by a stretch in the \(x\)-direction with scale factor \(\frac { 1 } { 2 }\) followed by a translation of \(\binom { 2 } { 10 }\). Find the equation of the new curve.
   \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-08_2716_38_143_2009}

1. a) Find

$$\int \frac { x } { x ^ { 2 } + 1 } d x$$ b) Find. $$\int 2 \pi ( 4 x + 3 ) ^ { 10 } d x$$ c) Find. $$\int \frac { 2 } { e ^ { 4 x } } d x$$