Course Content
第1章：行列式
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1.1 Determinant Expension 行列式的展开
1.2 Law of Determinants 行列式的性质
1.3 Cramer’s Rule 克兰姆法则
第二章：矩阵
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2.1 Addition, Subtraction and Multiplication of Matrices 矩阵的加减法和乘法
2.2 Inverse Matrices逆矩阵
2.3 Solve Linear Simultaneous Equations by Using Matrices 用矩阵解线性方程式
第七章：二项式定理
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7.1 Binomial Theorem with Natural Number Power 指数为自然数的二项式定理
7.2 General Term Formula for Binomial Expansion 二项展开式的通项公式
7.3 Binomial Theorem with Rational Power 指数为有理数的二项式定理
7.4 Application of Binomial Theorem in Approximate Calculation 二项式定理在近似值计算中的应用
第十二章：极限
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12.1 Concept of Limits 极限的概念
12.2 Limit of Functions 函数的极限
12.3 Limit of Infinities 无穷的极限
12.4 Limit of Series 数列的极限
12.5 Continuous Functions 连续函数
第十三章：微分法（一）
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13.1 Derivatives 导数
13.2 Continuous of Functions 函数的连续性
13.3 Differentiation Formulas 微分法则
13.4 Chain Rles链导法
13.5 Higher Degree Derivatives 高阶导数
13.6 Sandwich Theorem 挟挤定理
13.7 Derivatives of Trigonometric Functions 三角函数的导数公式
第十四章：微分法的应用（一）
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14.1 Tangent and Normal 切线与法线
14.2 The Increasing, decreasing & Extreme Values of Functions 函数的增减性&极值
14.3 Graph Sketching for Rational Functions 有理函数的作图
14.4 Optimization 最佳化
第十五章：不定积分（一）
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15.1 Fundamental Indefinite Integral 基本不定积分
15.2 Indefinite Integral of Trigonometric Functions 三角函数的不定积分
15.3 The Substitution Rules of Integration 换元积分法
第十六章：定积分及其应用（一）
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16.1 Fundamental Definite Integral 基本定积分
16.2 The Substitution Rules of Integration 换元积分法
16.3 Calculation of Area 面积的计算
16.4 Volume of Rotating Solid 旋转体的体积
高二数学 | 高级数学
About Lesson

15.3 The Substitution Rules of Integration 换元积分法

Evaluate the following indefinite integral
求下列不定积分

1) 

*** QuickLaTeX cannot compile formula:
\int{x(x^2+1)^\sfrac{3}{2}dx}

*** Error message:
Undefined control sequence \sfrac.
leading text: $\int{x(x^2+1)^\sfrac
Answer :
\frac{1}{5}{(x^2+1)}^\frac{5}{2}+C
2) 

*** QuickLaTeX cannot compile formula:
\int{x(x^2+1)^\sfrac{1}{3}dx}

*** Error message:
Undefined control sequence \sfrac.
leading text: $\int{x(x^2+1)^\sfrac
Answer :
\frac{3}{8}(x^2 + 1)^{\frac{4}{3}} + C
3) \int_{0}^{2}{x^2(1+2x^3)^3dx}
Answer :
3480
4) \int\frac{x^3}{\sqrt{1-x^4}}dx
Answer :
-\frac{1}{2}\sqrt{1-x^4}+C
5) \int\frac{(x+1)dx}{\sqrt[3]{x^2+2x}}
Answer :
\frac{3}{4} (X^2 + 2X)^{\frac{2}{3}} + C
6) \int{\frac{\sqrt{1-\sqrt x}}{\sqrt x}dx}
Answer :

7) \int{x\sqrt{x-1}dx}

Answer :
\frac{2}{5}\left(x-1\right)^\frac{5}{2}+\frac{2}{3}\left(x-1\right)^\frac{3}{2}+C
8) \int{x^3\left(x-3\right)^3\left(2x-3\right)dx}
Answer :
\frac{1}{4}x^4\left(x-3\right)^4+C
9) \int\frac{xdx}{\sqrt{2x+1}}
Answer :
\frac{1}{6}\left(2x+1\right)^\frac{3}{2}-\frac{1}{2}\left(2x+1\right)^\frac{1}{2}+C
10) \int{\frac{x^2}{\sqrt{1-x}}dx}
Answer :
-2\left(1-x\right)^\frac{1}{2}+\frac{4}{3}\left(1-x\right)^\frac{3}{2}-\frac{2}{5}\left(1-x\right)^\frac{5}{2}+C
11) \int{(x-1)\sqrt{x+1}dx}
Answer :
\frac{2}{5}\left(x+1\right)^\frac{5}{2}-\frac{4}{3}\left(x+1\right)^\frac{3}{2}+C
12) \int{(x-2)\sqrt{x-1}dx}
Answer :
\frac{2}{5}\left(x-1\right)^\frac{5}{2}-\frac{2}{3}\left(x-1\right)^\frac{3}{2}+C
13) \int{\frac{3x-1}{\sqrt{x+1}}dx}
Answer :
2(x + 1)^{\frac{3}{2}} - 8(x + 1)^{\frac{1}{2}} + C
14) \int{x(3x-2)^3dx}
Answer :
\frac{1}{45}(3x - 2)^5 + \frac{1}{18}(3x - 2)^4 + C
15) \int{xsin{(}2x^2)dx}
Answer :
-\frac{1}{4}\cos{2x^2+C}
16) \int{x{sec}^2{(}4x^2-5)dx}
Answer :
\frac{1}{8}\tan{(4x^2-5)}+C
17) \int{\frac{cos{(}1/t)}{t^2}dt}
Answer :
-sin\frac{1}{t} + C
18) \int{\frac{sin{\sqrt x}}{\sqrt x}dx}
Answer :
-2cos\sqrt {x} + C
19) \int{sin{x}{sec}^2{(}cos{x})dx}
Answer :
-tan(cos x) + C
20) \int{sin{x}cos{x}(sin{x}+cos{x})dx}
Answer :
\frac{1}{3}\sin^3{x}-\frac{1}{3}\cos^3{x\ }+C
21) \int{{sec}^3{x}tan{x}dx}
Answer :
\frac{1}{3}\sec^3{x}+C
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