Answer to: Evaluate the indefinite integral. Integral of sinh^2 x cosh x dx. By signing up, you'll get thousands of step-by-step solutions to your...
1.Identify the integral: We want to compute∫tanh(x)dx. 2.Recall the definition oftanh(x): The hyperbolic tangent function is defined as: wheresinh(x)=ex−e−x2andcosh(x)=ex+e−x2. 3.Use the identity fortanh(x): We can express the integral as: ...
Integration of Hyperbolic Functions: Techniques for integrating functions like sinh(x), cosh(x), etc. Linearity of the Integral: The principle that the integral of a sum is the sum of the integrals. Sequences and Series: For integrating functions represented as power series or when using techni...
sinhx=ex−e−x2 When we want to integrate a hyperbolic function, we usually use exponential form for the ease of calculations. Answer and Explanation: We will evaluate ∫sinhx2dx. Using the exponential form, sinhx2=ex2−e−x22 Therefore, {eq}\int{\sinh{x...Become...
Put x=3coshu,u≤0x=3coshu,u≤0, so dx=3sinh(u)dudx=3sinh(u)du Then I=∫3sinh(u2)9cosh2(u)du=∫tanh2(u)du=∫(1–sech2(u))du=u–tanh(u)+c=u−3sinh(u)3cosh(u)+c=cosh−1(x3)−x2–9−−−−√x+cI=∫3sinh(u2)9cosh2(u)du...
sinh(x)=∞∑n=0x2n+1(2n+1)! Then swapping the integral and summation signs, we get: I=∞∑n=01(2n+1)!⋅∫∞0x2n+2(1+cosh2(x))2dx But I am unable to proceed with the evaluation of the inner integral. Can someone help me evaluate the value of t...
x−∫01ln(1+x)1+x2 dx⏟x→1−x1+x=∫1∞ln(1+x+1+x2 )1+x2 dx−∫1∞lnx1+x2 dx⏟x→1x+πln28−∫01ln2−ln(1+x)1+x2 dx=12∫0∞ln(1+x+1+x2 )1+x2 dx⏟x→sinhx+...
Answer to: Evaluate the following definite integral. integral_1^2 1 / {x^3 + 5 x^2 + 6 x} dx By signing up, you'll get thousands of step-by-step...
{eq}\displaystyle \int_{\ln 9}^{\ln 36} e^{ \frac{x}{2} } dx {/eq} Definite Integrals: In mathematics, the definite integral is a similar process to calculating the indefinite integral of any function but a definite integral has the start and end values (the lower and the upp...
Evaluate: {eq}\int x \sec^2(x) \, \mathrm{d}x {/eq}. Integration: If F(x) is the primitive of f(x), then F(x) + c is also a primitive of f(x), where c is a constant. {eq}\int f(x) dx = F(x) +c {/eq} which is called the indefinite integral, where c ...