Step-by-step Solution

Calculate the integral of $\int\frac{\sin\left(x\right)^3}{\cos\left(x\right)}dx$

Go!
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Final Answer

$-\ln\left|\cos\left(x\right)\right|-\frac{1}{2}\sin\left(x\right)^2+C_0$

Step-by-step explanation

Problem to solve:

$\int\left(\frac{\sin\left(x\right)^3}{\cos\left(x\right)}\right)dx$
1

Rewrite the trigonometric function as the product of two lower exponents

$\int\frac{\sin\left(x\right)^{2}\sin\left(x\right)}{\cos\left(x\right)}dx$
2

Applying the trigonometric identity: $\sin^2(\theta)=1-\cos(\theta)^2$

$\int\frac{\left(1-\cos\left(x\right)^2\right)\sin\left(x\right)}{\cos\left(x\right)}dx$
3

Multiplying polynomials $\sin\left(x\right)$ and $1-\cos\left(x\right)^2$

$\int\frac{\sin\left(x\right)-\sin\left(x\right)\cos\left(x\right)^2}{\cos\left(x\right)}dx$
4

Split the fraction $\frac{\sin\left(x\right)-\sin\left(x\right)\cos\left(x\right)^2}{\cos\left(x\right)}$ inside the integral, in two terms with common denominator $\cos\left(x\right)$

$\int\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}+\frac{-\sin\left(x\right)\cos\left(x\right)^2}{\cos\left(x\right)}\right)dx$
5

Simplifying

$\int\frac{\sin\left(x\right)}{\cos\left(x\right)}dx+\int-\sin\left(x\right)\cos\left(x\right)dx$
6

The integral $\int\frac{\sin\left(x\right)}{\cos\left(x\right)}dx$ results in: $-\ln\left|\cos\left(x\right)\right|$

$-\ln\left|\cos\left(x\right)\right|$
7

The integral $\int-\sin\left(x\right)\cos\left(x\right)dx$ results in: $-\frac{1}{2}\sin\left(x\right)^2$

$-\frac{1}{2}\sin\left(x\right)^2$
8

Gather the results of all integrals

$-\ln\left|\cos\left(x\right)\right|-\frac{1}{2}\sin\left(x\right)^2$
9

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration

$-\ln\left|\cos\left(x\right)\right|-\frac{1}{2}\sin\left(x\right)^2+C_0$

Final Answer

$-\ln\left|\cos\left(x\right)\right|-\frac{1}{2}\sin\left(x\right)^2+C_0$

Problem Analysis

$\int\left(\frac{\sin\left(x\right)^3}{\cos\left(x\right)}\right)dx$

Main topic:

Calculus

Related formulas:

3. See formulas

Time to solve it:

~ 0.1 seconds