Chapter 14 Integral Calculus

Section Exercise

14.1 Indefinite Integral

14.1 Indefinite Integral Find each of the following indefinite integrals. $\displaystyle \int\left(3 x^{3}-4 \sqrt{x}+3\right) dx$ $\displaystyle \int\left(6 x^{2}-\frac{4}{x^{2}}\right) dx$ $\displaystyle \int\left(5-\frac{1}{\sqrt{x}}+\frac{1}{x^{3}}\right) dx$ $\displaystyle \int…

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Section Exercise

14.2 Finding the Constant of Integration

14.2 Finding the Constant of Integration Given that the gradient of a curve is $\displaystyle 2x^2 + 7x$ and that the curve passes through…

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Section Exercise

14.3 Integration of the form (ax + b)^n

14.3 Integration of the form (ax + b)^n Integrate each of the following with respect to $\displaystyle x$. $\displaystyle \int(2 x-3)^{4} dx$ $\displaystyle \int…

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14.4 Integration of Trigonometric Functions

14.4 Integration of Trigonometric Functions Find each of the following integrals. $\displaystyle \int 2 \left(\sin 3x + \frac{1}{2} \cos 2x\right) dx$ $\displaystyle \int \frac{3}{4…

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14.5 Integration of the Form 1/x and 1/(ax+b)

14.5 Integration of the Form 1/x and 1/(ax+b) Integrate each of the following with respect to $\displaystyle x$. $\displaystyle \frac{1}{2 x}$ $\displaystyle \frac{1}{2 x+3}$…

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14.6 Integration of Exponential Functions

14.6 Integration of Exponential Functions Find each of the following indefinite integrals. $\displaystyle \int e^{2 x+\pi} dx$ $\displaystyle \int e^{3-4 x} dx$ $\displaystyle \int…

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14.7 Definite Integral

14.7 Definite Integral Evaluate: $\displaystyle \int_{1}^{2} 3 x^{2} dx$ $\displaystyle \int_{1}^{3} \frac{4}{x^{3}} dx$ $\displaystyle \int_{-1}^{1}(2 x-3) dx$ $\displaystyle \int_{0}^{3}\left(10-x^{2}\right) dx$ $\displaystyle \int_{-1}^{2}\left(4 x^{2}-2 x\right)…

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14.8 Area Under a Curve

14.8 Area Under a Curve Find the area of each shaded region. Find the area between the curve with equation $\displaystyle y=f(x)$, the $\displaystyle…

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14.9 Area between Curves

14.9 Area between Curves Sketch the curve and find the total area bounded by the curve and the $\displaystyle x$-axis for each of these…

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Section Exercise

14.10 Volume Generated by Revolving Curves

14.10 Volume Generated by Revolving Curves Find the volume obtained when the shaded region is rotated through \( \displaystyle 360^\circ \) about the \(…

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14.11 Exam Style Problems (Pearson Edexcel)

14.11 Exam Style Problems (Pearson Edexcel) Figure shows part of the curve \( \displaystyle C \) with equation \( \displaystyle y=4-e^{2x} \).The curve \(…

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Section Exercise

14.12 Exam Style Problems (Cambridge)

14.12 Exam Style Problems (Cambridge) Do not use a calculator in this question. Show that $\displaystyle \frac{d}{dx}\left(\frac{e^{4x}}{4}-x e^{4x}\right)=p x e^{4x}$, where $\displaystyle p$ is…

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