积分算是微分的逆运算,积分可以用来计算曲线下的面积。多项式的类型不同,积分的公式也不同。
步骤
方法 1 的 2:
简单的积分
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1
大多数多项式适用的积分公式。 比如多项式:y = a*x^n.
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2
系数除以(n+1),然后指数加上1。 换句话说y = a*x^n 的积分是y = (a/n+1)*x^(n+1) .
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3
对于不定积分,一个多项式对应多个,所以要加上积分常数C。 因此本例的最终结果是y = (a/n+1)*x^(n+1) + C 。
考虑这样一个问题:在计算微分是,所有常数项都被省略。因此,在求积分时,积分结果可以加上任意的常数。
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4
根据这个公式,计算积分。 比如,y = 4x^3 + 5x^2 +3x 的积分是(4/4)x^4 + (5/3)*x^3 + (3/2)*x^2 + C = x^4 + (5/3)*x^3 + (3/2)*x^2 + C .
方法 2 的 2:
其他公式
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1
上文提到的公式不适用于x^-1或1/x的形式。 当你计算指数为-1的指数式的积分时,其结果是自然对数的形式。换句话说(x+3)^-1的积分是ln(x+3) + C 。
2
e^x的积分就是它自身。 e^(nx)的积分是1/n * e^(nx) + C ;因此,e^(4x) 的积分是1/4 * e^(4x) + C 。
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3
三角函数的积分需要记忆。 你要记住下面的积分公式:
cos(x) 的积分是sin(x) + C
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sin(x) 的积分是-cos(x) + C (note the negative sign!)
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根据这两个公式,你可以计算tan(x),即sin(x)/cos(x)的积分。 其积分是 -ln|cos x| + C ,你可以求它的微分看看。
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4
对于比较复杂的多项式,比如(3x-5)^4, 要使用替换法来求积分。 引入一个变量,比如u,来代替多项式,3x-5,这样可以简化所求的式子,然后套用上面的基本积分公式。
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5
计算相乘两函数的积分,使用分部积分法。